Lec 12 | MIT 6.042J Mathematics for Computer Science, Fall 2010


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visit MIT OpenCourseWare at ocw.mit.edu. Today we’re going to start
with a simple problem that many of you may
have already encountered. For example, when you
got a student loan or if your family took a
loan out on your house, you know, a home mortgage
loan, and it’s the problem of pricing an annuity. An annuity is a
financial instrument that pays a fixed
amount of money every year for some
number of years, and it has a value
associated with it. For example, with
a student loan, the value is the amount of
money they gave you to pay MIT, then later in life– every
year or every month– you’re going to send them a check. And you want to sort of equate
those two things to find out, are you getting enough
money for the money you’re going to pay back
monthly sometime in the future? So let’s define this, and
there’s a lot of variations on annuities, but we’ll
start with one that’s called an n-year $m-dollar
payment annuity, and it works by paying m dollars
at the start of each year, and it lasts for n years. Now, usually n is
finite, but not always, and in a few minutes
we’ll talk about the case when it’s infinite. But this includes
home mortgages, where you pay every
month for 30 years. Megabucks, the lottery. They don’t actually give
you the million dollars. They give you $50,000
per year for 20 years, and call it a million bucks. Retirement plans. You pay in every
year, and then you get some big lump sum later. Life insurance benefits,
you know, and so forth. Now, if you go to Wall
Street, this is a big deal. A lot of the stuff that
happens on Wall Street involves annuities in
one form or another, packaging them all up
and, in fact, we’ll look at later when
we do probability. It was how these things
were packaged and sold that led to the sub-prime
mortgage disaster, and we’ll see how some confusion
over independents, when you look at random variables,
led to the global recession, a real disaster. Of course, some people
understood how all that worked, made hundreds of billions
of dollars at the same time. So it was sort of money went
from one place to another. So it’s pretty
important to understand how much this is worth. What is this
instrument– a piece of paper that says
it will pay you m dollars at the beginning of
each year for n years, what is that worth today? For example, say I
gave you a choice– the Megabucks choice–
$50,000 a year for 20 years or a
million dollars today. How many people would prefer
the $50,000 a year for 20 years? A few of you. How many would prefer the
million bucks right up front? Much better. OK. Always better to have
the cash in hand, because there’s things
like inflation– pretty low now– interest. You can put the
money in the bank or invest it and make
some money hopefully. So the million dollars
today is a lot better, which is why the State pays you
50 grand a year for 20 years. It’s better for them, and
they call it a million bucks. So that was pretty
clear, but what if I gave you this option– 700
grand today or 50 grand a year for 20 years? How many people want the cash
upfront– 700 grand only. A few. How many people want 50
grand a year for 20 years? All right, we’re almost–
that’s pretty close to half way. How about 500 grand today versus
50 grand a year for 20 years? How many want half
a million today? A lot of people like the cash. You know, it’s this kind of a
time, you have the recession, it’s a disaster on Wall Street. You know, Street Wall
Street didn’t like the cash. How many want, instead
of a half a million up front, 50 grand
a year for 20 years? All right, now
we’re about halfway. All right, well
that’s pretty good. So we’re going to find
out what you should pay, or at least one way
of estimating that. Now, to do that, we’ve
got to figure out what $1 today is worth in a year. And to do that, we
make an assumption, and the assumption
is that there’s a fixed– we’ll call
it an interest rate. It’s sort of the devaluation
of the money per year. And we’re going to call it p. Later we’ll plug
in values for p, but you can think of
it as like 6%, 1%. You know, it’s the money,
if you put money in a bank, they’ll give you some
percent back every year. And, of course, the fact
that different people have different ideas
of what this would be, allows people to make
money on Wall Street. As we’ll see, a
slight difference in p can make big differences in
what the annuity is worth. So for example, $1 today
is going to equal 1 plus p dollars in one year. Similarly, $1 today–
how much is that going to be worth in two years? Say that p stays fixed,
the same over all time. One plus p squared,
because every year you multiply what you got by
1 plus p, because that’s the interest you’re getting. All right, we’ll think of
it in terms of interest. In three years, $1 today
is worth 1 plus p cubed in three years and so forth. All right. Now, what we really care about
is what’s $1, or m dollars, worth today if you’re
getting it next year? So we need to sort of flip
this back the other way. So what is $1 in a year
worth today in terms of p? So if I’m going to
be paid $1 in a year, what would be the equivalent
amount to be paid today? One over 1 plus p, because
what’s happening here is, as you go forward in a year,
you just multiply by 1 plus p. So 1 over 1 plus p turns
into $1 in a year– being paid in a year. All right. What is $1 a year in
two years worth today? One over 1 plus p squared. So $1 in two years is
worth this much today. Well, now we can use
this to go figure out the current value
of that annuity. We just figure out what every
payment is worth today and then add it up. So we’ll put the
payments over here, and we’ll compute
the current value of every payment on this side. So with the annuity,
the way we’ve set it up is it pays n dollars
at the start of every year, so the first payment is now. So the first of the
n payments is now, and since it’s being paid
now, that’s worth m dollars. There’s no devaluation. The next payment is m
dollars in one year, and so that’s going to be
worth m over 1 plus p today. And the next payment is
m dollars in two years. That’s worth m over
1 plus p squared, and we keep on going
until the last payment. It’s the n-th payment, so it’s
m dollars in n minus 1 years. And so that’s going to
be worth m over 1 plus p to the n minus 1. All right. So we can compute the current
value of all those payments, then the annuity is computed–
the value is computed just by adding these up, of
all the current values. So the total current value is
the sum i equals 0 to n minus 1 of m over 1 plus p to the i. And that is the
total current value. That’s what you should
pay today for the annuity. Any questions? What we did here? All right. Well, of course, what we’d like
is a closed form expression here. Something that’s simple
so we could actually get a feel without having
to add up all those terms, and that’s not hard to get. In fact, let’s put
this sum in a form that might be more familiar. This equals– we’ll pull
the m out in front– and let’s use x to be 1
over 1 plus p to the i. And so x equals 1 over 1 plus
p, and I wrote it this way because this might be familiar. Does everybody remember
that from– I think it was the second recitation? Anybody remember the formula? What this evaluates to? The sum of x to the i, where
i goes from 0 to n minus 1? Remember that? One minus x to the n. Remember 1 minus x. In the second
recitation, I think, we proved that this equals that. What was the proof
technique we used? Induction. OK? So, in fact, there’s
a theorem here. For all n bigger and equal
to 1 and x not equal to 1, we proved the sum from i equals
0 to n minus 1 x to the i equals 1 minus x to
the n over 1 minus x. And so this is a
nice, closed form. No sum any more, just
that, which is nice. Now, induction proved
it was the right answer. Once you knew it– we gave
it to you– using induction to prove that
theorem wasn’t hard. What we’re going to look at
doing this week and next week is figuring out how
to figure out this was the answer in
the first place. Methods for doing that– to
evaluate the sum– and there’s a lot of ways that you can
do that particular sum. Probably the easiest is known
as the perturbation method. This sometimes works. Certainly with sums like
that, it often works. The idea is as follows. We’re trying to compute
the sum S, which is 1 plus x plus x squared
plus x to the n minus 1, and what we’re going to do
is perturb it a little bit and then subtract to
get big cancellation. In this case, it’s
pretty simple. We multiply the sum by x to get
x plus x squared plus– I’ve defined S to be that,
x times S– well, I get x plus x squared and
so forth, up to x to the n, and now I can subtract one from
the other and almost everything cancels. So I get 1 minus x
times S equals 1. These cancel, cancel, cancel. Minus x to the n. And therefore S equals 1 minus
x to the n over 1 minus x. So that’s a vague method. This gets used all the time
in applied mathematics, and they call it the
perturbation method. Take your sum, wiggle
it around a little bit, get something that looks
close, subtract it, everything cancels,
life is nice, and all of a sudden you’ve
figured out the answer. So getting back to
our annuity problem, we can plug that
formula back in here. So the value of the annuity
is m times 1 minus x to the n over 1 minus x. We’ll plug in x equals
1 over 1 plus p, and we get m 1 minus
1 over 1 plus p to the n over 1 minus 1 over
1 plus p, just plugging in. And now to simplify this, I’ll
multiply the top and bottom by 1 plus p, and I’ll get 1
plus p minus 1 on the bottom. Just gives me a p on the bottom. I have a 1 plus p on
the top minus 1 over 1 plus p to the n
minus 1, all over p. So now we have a formula–
closed form expression formula– for the
value of the annuity. All’s we’ve got to plug in
is m, the payment every year, n, the number of years, and
then p, the interest rate. And so, for example, if
we made m be $50,000, as in the lottery. We made n be 20 years, and
say we took 6% interest, which is actually very good these
days, and I plug those in here, the value is going
to be $607,906. All right. So those of you that preferred
700 grand– if you assume 6% interest– you’re right. Those of you who
preferred 500 grand, no, you’re better off waiting and
getting your 50 grand a year. Now, of course, if the
interest rate is lower, well, that changes things. That shifts it even more. The annuity is worth even more
if the interest rate is lower– if p is smaller. In fact, say p was 0. Say the interest rate is 0, so
$1 today equals $1 tomorrow, then what is the lottery worth? A million dollars. And the bigger p gets, the
less your payment is worth. Any questions about that? OK. What if you were paid $50,000 a
year forever– you live forever or it goes to your
estate and your heirs, $50,000 a year forever or
a million dollars today. How many people want the
million dollars today? How many want 50
grand a year forever? Sounds good. You know, that’s an infinite
amount of money, sort of. It’s not as good as it sounds. Let’s see why. So this is a case where
n equals infinity, and so I’ll claim that
if n equals infinity, then the value of this
annuity is just m times 1 plus p over p. Let’s see why that’s the case. You know, it sounds
hard to evaluate, because it’s an infinite
number of payments, but what happens here
when n goes to infinity? What happens to this thing? That goes to 0 as
n goes to infinity, as long as p is bigger than 0. So we’re going to
assume 6% interest. So that goes away, so
the annuity is worth just that, m times 1 plus p over
p, because the limit as n goes to infinity of 1 over
1 plus p to the n minus 1, that’s going to 0. So the value for m is $50,000,
and at 6% V is only $883,000. So you’re better off
taking a million dollars today than $50,000
a year forever. Now, if you think about
it, and think about it as an interest
rate, why should it be obvious that you’re better
off with a million dollars today than 50 grand
a year forever? Think about what you could
do with that million dollars if you had it today
at 6% interest. What would you do with it to
make more money than 50 grand a year forever? Yeah. AUDIENCE: You could have like–
you could make $50,000 per year just off of [INAUDIBLE]. PROFESSOR: Yeah. In this model, if the
interest rate is 6%, you can put in the bank. It makes 6% every year, that’s
60 grand a year forever. Better than 50 grand
a year forever. So maybe it’s– even
without doing the math, you can tell which
way it’s going to go, but this tells you
exactly what it’s worth. Any questions? OK. Yeah. AUDIENCE: [INAUDIBLE]
current value– that’s how much it’s
worth to you right now. PROFESSOR: Yeah. AUDIENCE: So and, like, if
you have $1 in the future– PROFESSOR: Yeah. AUDIENCE: –where S right
now is m over 1 plus p– PROFESSOR: Yeah. AUDIENCE: –is it worth less to
you [INAUDIBLE] then later on, it’s going to be
worth more to you. PROFESSOR: Ah, so
if you move yourself forward in time, $1 a year, in
a year it’ll worth $1 to you– AUDIENCE: Yeah. PROFESSOR: –but today it’s
worth less than $1 to you, because you could take the
dollar today and invest in the bank, and it’s worth more
in a year, because the money grows in value, as a
way to think of it, because you can
earn interest on it. What’s that? AUDIENCE: You could spend it. PROFESSOR: Yeah. Yeah, if you just
spend it, well, then at least you
had the use of what you spent it on for the year. So there’s some other
kind of value, right? Maybe you bought a
house or something that– maybe something that
even appreciated in value. OK. But these things
get sort of squishy, and that is where
a lot of people make money on Wall Street, is
because different companies have different needs for money. They have different views of
what the interest rates are going to be, and you
can play in the middle and make a lot of
money that way. So more generally, there’s
a corollary to the theorem, and that is that if the absolute
value of x is less than 1, then the sum i equals 0
to infinity x to the i is just 1 over 1 minus x. We didn’t prove this back
in the second recitation, because there’s no
n to induct on here, but the proof is simple
from the theorem. And it’s simply because
if x is less than 1, an absolute value, as n goes
to infinity, that goes to 0, and so you’re just left
with 1 over 1 minus x. So, for example,
what’s this sum? This one you all know, I’m sure. Out to infinity. What’s that sum to? To 2. Yeah. It’s 1 over 1 minus
1/2, which is 2. What about this sum
out to infinity? What does that sum to? AUDIENCE: [INAUDIBLE] PROFESSOR: Yeah, 3/2, 1
over 1 minus 1/3 is 3/2. So, easy corollaries. These are all examples
of geometric series. That’s what a definition
of a geometric series is. Something that’s going
down by a fixed– each term goes down by the same
fixed amount every time. And geometric
series, generally sum to something that is very
close to the largest term. In this case, it’s 1. Very common, because
of that formula. It’s 1 over 1 minus x. Any questions about this
or geometric series? All right. Well, those are
straight geometric sums. Sometimes you run into things
that are a little bit more complicated. For example, say I have this
kind of a sum, i equals 1 to n, i times x to the i. Now, those are adding up x
plus 2x squared plus 3x cubed, and so forth, up
to n x to the n. You know, that’s a
little more complicated. The terms are
getting– decreasing by a factor of x, increasing by
1 in terms of the coefficient every time. A little trickier. So say we wanted to get a
closed form expression for that? There are several
ways we can do it. The first would be to try to use
perturbation– the perturbation method. Let’s try that. So we write S equals x plus 2x
squared plus 3x cubed plus nx to the n, and let’s try
the same perturbation. Multiply by x, I
get that x squared plus 2x cubed plus n minus 1x to
the n, plus nx to the n plus 1. And then I subtract to try
to get all the cancellation. So then I do that. I get 1 minus x times S. Well, I
didn’t quite cancel everything. X plus x squared
plus x cubed plus x to the n plus n– or
minus nx to the n plus 1. Ah, it didn’t quite work. Anybody see a way that
I can fix this up? What about this piece? That’s still a mess here. Can I simplify that? Yeah? AUDIENCE: [INAUDIBLE] PROFESSOR: Yeah. There’s a simpler way. Yeah. AUDIENCE: That’s a
geometric series. PROFESSOR: That’s
a geometric series. We just got the formula
for it, so that’s easy. This equals 1 minus x to the
n over 1 minus x minus the 1, because I’m missing the 1 here. So we can rewrite this
whole thing over here. Oops. Yikes. Got attacked. What’s that? AUDIENCE: Would it be 1
minus x should be n plus 1? PROFESSOR: Yes it would. That’s right. I’ve got to add 1 to there. OK. So that’s good. So that says that
1 minus x times S equals 1 minus x to the n
plus 1 over 1 minus x minus 1, and then I’ve got to remember
to subtract that term too. Minus nx to the n plus 1. That means now I
just divide through, and I simplify– divide through
by 1 minus x– and simplify, and I get the following formula. I won’t go through all the
details, but it’s not hard. Let’s see if I got that right. Yeah, that looks right. OK, so that is the
closed form expression for that sum, which we can
get from the perturbation method and the fact
that we’d already done the geometric series. There’s another way to
compute these kinds of sums, which I want to show you,
because it can be useful. So we’re going to do the same
sum and derive the formula a different way. This method is called
the derivative method, and the idea is to start
with a geometric series which it’s close to and then
take a derivative. So for x not equal to 1, we
already know from the theorem that i equals 0 to n, x to the
i equals 1 minus x to the n plus 1 over 1 minus x. That was the theorem. We already know that. Now, I can take the
derivative of both sides, and let’s see what we get
by taking the derivative. Well, here I get the sum. The derivative of x to the
i is just i times x to the i minus 1. The derivative over here
is a little messier. I’ve got to have–
well, I take 1 minus x times a derivative of that. Derivative of this is now
minus n plus 1, x to the n. Then I take this times
the derivative of that is now minus minus 1, 1
minus x to the n plus 1, and then I divide
by that squared. Now, when we compute all that
out, we get this, 1 minus n plus 1, x to the n plus
nx to the n plus 1 over 1 minus x squared. I won’t drag you
through the algebra, but it’s not hard to
go from there to there. This is pretty close
to what we wanted. We’re trying to figure out
this, and we almost got there. What do I do to finish it up? AUDIENCE: Multiply by x. PROFESSOR: Multiply by x. Good. So if I take this and multiply
by x, i equals zero to n, ix to the i equals x minus
n plus 1, x to the n plus 1, plus nx to the n plus 2,
all over 1 minus x squared. OK? Which should be the same–
yeah– the same formula we had up there. So that’s called the
derivative method. You can start manipulating–
you treat these things as polynomials– these sums–
and you start manipulating them like you would polynomials. In fact, there’s a whole
branch of mathematics called generating functions
that we won’t have time to do in this class that’s
in chapter 12 of the text. But you do things
like that to get sums. Any questions about
what we did there? You can also do a version where
you take integrals of this if you want, and
then you get the i’s in the denominator instead
of those coefficients. For homework, I
think we’ve given you the sum of i squared x to the i. How do you think you’re
going to do that? Any thoughts about how
you’re going to solve that? Get the sum, a closed
form for the sum of i squared x to the i? AUDIENCE: Do the
derivative method twice. PROFESSOR: Yeah, do it twice. Take this, which now you know. Take the derivative again. Won’t be too hard. You can also take the version of
this where n goes to infinity. Let’s do that. If the absolute value of x
is less than 1, the sum i equals 1 to infinity of ix to
the i, what does that equal? This one, you can see
it easier up here. What happens when
n goes to infinity? What does this do? X is less than 1,
an absolute value. What happens to this
as n goes to infinity? This term. Goes to 0, right? This gets big, but this
gets smaller faster. What happens to this term
as n goes to infinity? Same thing, 0. All I’m left with is x
over 1 minus x squared. Now, this formula
is useful if you’re trying to, say, get
the value of a company, and the company is growing. Every year the company grows
its bottom line by m dollars. So the first year, the
company generates m dollars, the next year it generates
two m dollars in profit, the next year is
three m dollars. So you’ve got an
entity that every year is growing by a fixed amount. It’s not doubling every
year, but every year adds in m dollars
more of profit. What would you pay
to buy that company? What is that worth? So you can think of this
as, again, an annuity. Here the annuity pays im
dollars, in this case, at the end, not the beginning,
say, of the year i forever. This company is– or this
annuity– is worth, well, we just plug into the formula. Instead of $1 each year, it’s
m so there’s an m out front. x is 1 over 1 plus p, and
then we have 1 minus 1 over 1 plus p squared. And if we multiply the top and
bottom by 1 plus p squared, we get m 1 plus
p over p squared. So it’s possible with
a very simple formula to figure out how
much you should spend to buy this company,
what its value is today. So, for example, say the company
was adding $50,000 a year in profit. The interest rate was 6%,
the value of this company is $14 million– $14.7
million, just plugging into that formula. So people that buy
companies and stuff, they use formulas like this
to figure out what it’s worth. Of course, you’ve
got to make sure it’s really going to keep paying
the $50,000 more every year and that this is the
right interest rate to be thinking about. You know, and the
guys on Wall Street– the bankers on Wall
Street– they all have their estimations for
what these things are– the value of p they would
put into these formulas. Any questions about that? Yeah. AUDIENCE: [INAUDIBLE] PROFESSOR: OK. Good. So this one is OK? OK. I plugged x equals
1 over 1 plus p, like we did before–
remember for the annuity– because every year
you’re degrading it, devaluing by 1 over 1 plus p. So that’s the x term, and it’s
paying– in the i-th year, it’s paying im dollars. All right? So the first year it pays n
dollars, the next year 2 m, the next year 3 m,
the next year 4 m, but every year you’re
knocking it down by 1 plus 1 over p to the number of years. So what you get– the sum
you’ve really got here– is i equals 1 to infinity,
im dollars are paid, but those dollars are
worth 1 over 1 plus p to the i today,
the current value. That’s a good question. I should’ve said that. That’s a great question. So that’s how we
connected this up, because you’re getting paid
this much in our years, and that’s worth
that much degradation or that much devaluation
today, and now we add up a total current value. So even a company that is
paying you more and more every year still has a finite
value, because the extra– the payments are increasing
but only linearly. The value today is
decreasing geometrically, and the geometric decrease
wipes out the value of the company in the future. Yeah. AUDIENCE: Are you [? squaring ?]
quantity of [INAUDIBLE]. PROFESSOR: What did I do? Oh, wait, wait, wait. I screwed up here too. Is that what you’re
asking about? Yeah. That’s what I should
have done, right? Because I got 1 minus x is
1 minus 1 over 1 plus p. That gets squared, and now when
I multiply 1 plus p squared, it’s multiplying
this by 1 plus p. It’s 1 plus p minus 1 is p. So I have p squared. All right, so this part’s OK. That part I wrote wrong. That’s good. Any other questions? So let’s do a simple example. What is this sum? A 1/2 plus 2/4 plus
3/8 plus 4/16 forever. What’s that sum equal? You can plug that
in the formula. AUDIENCE: [INAUDIBLE] PROFESSOR: Yeah. That’s right. Good. These details. Otherwise that
sum would be what? If I didn’t put
the negative here, it’s going to be infinity,
and that’s not so interesting. The negative makes it more
interesting, so I got 1 over 2 to the i in there. What’s it worth then? Well, I can plug in the formula. What’s x? One half. So I get 1/2 over 1 minus
1/2 squared is 1/2 over 1/4, and that’s 2. Any questions on that formula? It’s amazing how useful
these things get to be later. So that’s sort of the
geometric kinds of things. Next I want to talk about more
of the arithmetic kinds of sums and what you do there. In fact, we’ve already
seen one that we’ve done. If I sum i equals 1 to n of
i– I think we’ve already done this one– that’s just
n times n plus 1 over 2, and probably most
of you even learned that formula back
in middle school, I’m guessing– maybe before. How many people know
the answer for this sum? The sum of the squares–
the first n squares. Somebody knows. What is it? AUDIENCE: n times n plus 1
times 2n plus 1, all over 6. PROFESSOR: Very good. That is correct. Most people don’t
remember that one. It’s a little harder to derive. How would you prove
this by induction? Unfortunately, induction
doesn’t tell you how to remember what
the formula was, and there’s a couple of
ways you can go about that. One is, you can
remember or guess that the answer is
a polynomial in n. In fact, because
you’re summing squares, you might guess that it’s
a cubic polynomial in n, and if you remember just that
or guess just that, then you could actually plug in
values and get the answer. And this is– you
know, a common method of solving these sums
is you sort of guess the form of the solution. In this case you might
guess that for all n, the sum i equals 1 to n of
i squared equals a cubic. And then what you would do is
plug in the value n equals 1, n equals 2, maybe
even– we’ll make it n equals 0– make
it simple and start getting some constraints
on the coefficients. If you would plug in n
equals 0, the sum is 0. The polynomial evaluates to d. That tells you what d has got
to be right away. n equals 1. The sum is 1, and when you
plug into the polynomial, you get a plus b plus c plus d. n equals 2. Well, that’s 1 plus
4 is 5, 2 cubed is 8, so you have 8a plus
4b plus 2c plus d, and you’ll need one more since
you’ve got four variables. Let’s see, 1 plus
4 plus 9 is 14. I’ve now got 3 cubed is
27a plus 9b plus 3c plus d. So now I’ve got four
equations and four variables and, with any luck, I can
solve that system of equations and get the answer. And, in fact, you can. When you solve this system, you
get a equals 1/3, b equals 1/2, c equals 1/6, and d equals 0. And that’s exactly what
you get in that formula. So that’s a way to reproduce
the formula if you forgot it. Now, this method–
really to be sure you got the right answer–
you’ve got to go prove it by induction, because I
derived the answer– if it was a polynomial, I would
have gotten it right, but I might be
wrong in my guess. And to make sure
your guess is right, you’ve got to go back
and use induction to prove it for this approach. Yeah. AUDIENCE: How do you know that
it would be [INAUDIBLE] and not some higher power? PROFESSOR: Well, it turns
out that anytime you’re summing powers, the answer
is a polynomial to one higher degree. So if you just remembered that
fact, or you guessed that fact. Another way to sort of
imagine that might be true is that I’m getting
n of them, so I might be multiplying–
the top one is n squared, so I’m going to have n
of them about n squared. Might be something like n cubed. That’s another way you could
think of it, to guess that. Any other questions on this? So far, all these sums had nice
closed forms, and a lot of them do that you’ll encounter
later on, but not all, and sometimes you get sums that
don’t have a nice closed form– at least nobody has
ever figured out one, and probably doesn’t
always exist. For example, what if I
want to sum the first n square roots of integers? Let’s write that down. So say I want a closed
form for this guy. Nobody knows an answer
for that, but there are ways of getting very
good, close bounds on it that are closed form, and
these are very important, and we’re going to use
this the rest of today and the rest of next time. And they’re based on replacing
the sum with an integral, and the integral is very
close to the right answer, and then we can see what
the error terms are. So let’s first look
at the case when we’ve got a sum where the
terms are increasing as i grows, and we’ll call
these integration bounds, and a general sum will look
like this– i equals 1 to n of f of i, and the first case is
when f is a positive increasing function, increasing in i. Integration bounds, and so
we’re increasing function. So let me draw a picture
that will hopefully make the bounds that we’re
going to get pretty easy. So let’s draw the
sum here as follows. I’ve got 0, 1, 2, 3, n
minus 2, n minus 1, n, and draw the values of f here. Here’s f of 1, f of 2–
it’s increasing– f of 3, f of n minus 1, and f of n. Then I’ll draw the
rectangles here. So this has area of f of
1, this has area f of 2, this has area f of 3,
and we keep on going. Let’s see, this will
be f of n minus 2 on this one– I’ll just do f of
n minus 1, draw this guy here. So its unit width, its
height is f of n minus 1, so its area is f of n
minus 1, and then f of n. And let me also–
so the sum of f of i is the areas in the rectangles. That’s what the
sum is, and I want to get bounds on this
sum using the integral, because integrals are
easier to compute. So let’s draw the function
f of x from 1 to n. All right, so this is
f of x as a function. Now I claim that the sum
i equals 1 to n of f of i is at least f of 1 plus
the integral from 1 to n, f of x, dx. Now, the integral
from 1 to n of f of x is this stuff, the
stuff under the curve. It comes down here,
starts at 1, and it’s the stuff under the curve. And what I’m saying
here is that if you take that stuff under
the curve and add f of 1, which is
this piece, that’s a lower bound on our sum. The sum’s bigger than that. So what I’m saying is the
area in the rectangles is at least as big as the
area in the first rectangle plus the area under the curve. Does everybody see why that is? I’m saying the sum is the
area in the rectangles, right? That’s pretty clear. And that is at least as big
as the first rectangle f of 1 plus the stuff under the
curve, which is the integral, and I’ve left– I’ve
chopped off these guys. That’s extra. OK? Is that all right? So lower bound. Any questions on
the lower bound? This is a picture proof, which
we always tell you not to do, but we’re going to do one here. And, of course, it totally hides
why did I need f is increasing, but we’ll see that in a minute. The proof would not work
unless it is increasing here. Any questions,
because now going I’m going to do the other
bound, the other side. I also claim– this will be
a little trickier to see– that the sum is at most f of n
plus the integral from 1 to n. So this is the lower
bound add in f of 1, the upper bound
just add in f of n. So let’s see why that’s true. Now, to see that,
this is– I’m not going to be able to draw it. I want you to imagine taking
this curve and the area under it, down to here, and
sliding it left one unit. sliding it left to here, sliding
it left one unit over to here. Now, when I slide it left one
unit, did the area under it change? No. It’s the same area
under it, just where it sits on the picture
is now out here. It’s this area under this
guy, but it’s the same thing, its the same integral. And you can see that
it’s more than what’s in these rectangles, because
I got all this stuff. And, of course, I didn’t
even include this, so now I add the f n. So if I take the area under the
curve, which is the integral, shift it left one,
so it only goes up to here now, and then add in
this rectangle, that dominates the area in the rectangles. Bigger than. Do you see that? I could do a lot of
equations on the board but, for sure, that would
be hopeless to follow. Any questions about this? Yeah. AUDIENCE: I guess I understand
the lower amounts because we’re cutting off the triangles. PROFESSOR: Yeah. But is there– is it
a lot of hand waving, or am I just missing something,
that it’s always going to be the f of n is what we– PROFESSOR: Yeah. There’s a little
hand waving going on, but I do believe it is true. With equations you
can make it precise. Let’s look at it again,
do this one more time. AUDIENCE: [INAUDIBLE] PROFESSOR: Yeah, I’m
waving hands a little bit, but it also– hopefully
the intuition comes across, because when I
shift left by one, this point becomes this
point, and that point becomes that point, and
I’m catching sort of the cusp of these rectangles,
and now I take this curve here, shifting the whole
curve left one unit, and the area doesn’t change. It would be equivalent of
taking the integral from 0 to n minus 1. You notice what I’m
doing– of f of x plus 1. There’s another way to
look at it mathematically, maybe– probably the
formal way to do it– but the idea is that
this now contains all these first n
minus 1 rectangles, are contained underneath it. And then I just add this
in, and I’m good to go. I’ve contained all
the rectangles now for the upper bounds. All right,
mathematically what this is, this curve is
0 to n minus 1, fx plus 1, which is the
same thing as what that is. Any questions for that? But now I have good
bounds on the sum. We know that the sum is at
least this and at most that, and those two bounds only differ
by a single term in the sum, so they’re very close. These actually are good
formulas to write down for your crib sheet. That is worth doing on the test,
and we’ll use them more today, and we’ll also use
them on Thursday. So now we can actually
get close bounds on the sum of the square roots. So let’s see how
this works for f of i equal to the square root
of i, which is increasing. So I compute the integral from
1 to n of square root of x dx, and that’s just x to the 3/2
over 3/2, evaluated at n and 1, and that just equals 2/3
n to the 3/2 minus 1. And now I can compute
the bounds on the sum of the first n square roots. So I know that square root
i equals 1 to n– well, the upper bound is f
n plus the integral, and the lower bound is f
of 1 plus the integral. What is f of n? That’s the square root of n. What is f of 1? The square root
of 1, which is 1. So I’ll plug that in here. I get 2/3 n to the 3/2 plus
1 minus 2/3 is plus 1/3, and here I get 2/3 n to the
3/2 plus the square root of n minus 2/3. So now I have pretty good
bounds on the sum of the first n square roots using this method. So for example,
take n equals 100. This number evaluates to 667. This number evaluates to
676, and the difference is 9, which is the square
root of 100 minus 1. This is the square
root of n, this is 1, so the gap here, square root
of n minus 1, and n equals 100. Square root of 100 minus
1 is 9, so I shouldn’t be surprised my gap here is nine. So I didn’t get exactly
the right answer, but I’m pretty close here. Now, as n grows, what happens to
the gap between the upper bound and the lower bound? What does it do? It gets bigger. So my gap gets bigger. That’s not so nice. Doesn’t always stay 9
forever, but somehow though this is still pretty
good, because the gap grows but the gap only grows
as square root of n, where the answer– the bounds–
are growing as n to the 3/2. In other words, my error
is somewhere around here, and that gets smaller
compared to my answer, which is somewhere around there. And there’s a special
notation that people use. In fact, let’s write that
down and then do the notation. Another way of writing this is
that the sum i equals 1 to n of square root of i. The leading term here
is 2/3 n to the 3/2, and then there’s some error
term– delta term here. We’ll call that
delta n, and we know that the error term is at
least 1/3 and, at most, the square root of n minus 2/3. So this delta term is bound
by the square root of n. That’s getting
bigger as n gets big, but this value compared to
your answer is getting small. That’s nice, and so the way that
gets represented is as follows. We would say– it’s using
that tilde notation. We write tilde 2/3
times n to the 3/2, and now we’ve gotten
rid of the delta, because this tilde is
telling us that everything else out here gets small
compared to this as n gets big. And the formal definition–
Let’s write out the formal definition for it. Now, a lot of times
you’ll see people use this symbol to mean about. That’s informal. When I’m using it here,
it’s a very formal meaning mathematically. A function g of x is
tilde, a function h of x means that the limit as x goes
to infinity of g over h is 1. In other words, that
as x goes to infinity– as x gets big– the ratio
of these guys becomes 1. And let’s see if
that’s true here. Well, square root
of i equals this, so I need to show the limit
of this over that is 1. So let’s check that. A limit as n goes to
infinity of 2/3 n to the 3/2 plus that delta term
over 2/3 n to the 3/2. Well, that equals– divide out
by 2/3 n to the 3/2, I get a 1. Did I– should I
have subtracted? No, that’s OK. One plus delta n over
2/3 n to the 3/2. If I can pull the 1 out front,
that’s 1 plus the limit, and this is now delta n is
at most square root of n, so I get square root of
n over 2/3 n to the 3/2. Square root of n over n to
the 3/2, that goes to 0. So this equals 1. So this limit is 1,
and so therefore I can say that the sum of
the first n square roots is tilde 2/3 n to the 3/2. Any questions about this? We’re going to do a lot of this
kind of notation next time. Yeah. AUDIENCE: When you
took the integral and got 2/3 into
the 3/2 minus 1, why did the minus 1 not become
part of the actual solution and become part of the delta? PROFESSOR: OK. So you brought it
up to here, right? OK. So then I plug that
into the integral that appears on both sides here,
and here I add the f 1, here I add f n, and now I
have the lower bound here and the upper bound here. Are you good with those? All right. Now some judgment takes
place, and what I’m really trying to do here is figure out
what are the important terms in these bounds as n gets big? How big is this
growing as n gets big? Well, as n gets big, the
1/3 is not doing much. As n gets big, the
square root of n grows, but it’s nothing
like what’s happening here. If you had to
describe to somebody what’s going on in this bound,
would you start here or here? No. You’d start here. This is the action, and there’s
a little bit– the rest is just in the slop, in the air. And so now I’ve used judgement
to say that delta is somewhere in this stuff here. What’s really happening, and
the nice thing is they match. The lower bound and the upper
bound match on that term. So what I do is I write
it equals this term plus something that’s
smaller and, in particular, it’s between 1/3 and
square root of n minus 2/3. All right. So what I’m trying
to capture here is just the guts of what’s
happening to this function as n grows, and the
guts of it is this. It’s not exactly equal to
2/3 times n to the 3/2, but it’s close
and, in fact, if I take the limit of this over
that, that limit goes to 1. It’s a way of saying they’re
approximately the same that’s called asymptotically
the same, and we’ll talk a lot more about
asymptotic notation next time. We’ll give you five more symbols
besides tilde that people use. Any questions about– maybe
start with the bounds. Any question on the
bounds that we got? That’s the integration method,
first getting the bounds. You take the integral, you add
f of 1 for the lower bound, you add f of n for
the upper bound, and it’s in between,
somewhere in there. Questions there? All right. Then we plugged
it in, and now we look at this tilde notation
that says– well, first we’d write it like this. The sum is this value plus an
error term and, lo and behold, that error term is small. If I take the limit of the whole
thing divided by the big term, I get 1, which means this
thing is really not important. So I write this. Questions on that? All right. There’s one more
case to consider, and now we’re going to go back
to the integration bounds, and that is when f is
a decreasing function, and we’re going to do
the analysis for that, and then we’ll be all done. So we’re going to look
at integration bounds when f is decreasing
and positive still. The example here might
be, for example, the sum i equals 1 to n, 1 over
the square root of i. Say you had to get some idea
of how fast that function is growing as a function of n. I’m summing the first n
inverse as the square roots. What is that
roughly going to be? How fast does that grow
as a function of n? So let’s do that. And, of course, 1
over square root of i decreases as i gets bigger. So let’s do the general picture
again and see what happens. So we have 0, 1, 2, 3, n minus
2, n minus 1, n, and now f of n is the small term and f of n
minus 1, f of 3 here, f of 1. Now, I’m going to
draw the rectangle. This has area f of 1. This one has area f of 2. This one has area f of
3, and then this one has area f of n minus 1 here,
and then, lastly, area f of n. So the sum is the area in the
rectangles, just like before, except now the rectangles
are getting smaller. Let’s draw the integral
like we did before. The integral is the area
under this curve, f of x here, just like before. So this is f of x,
only it’s decreasing. Now, let’s take the area under
this curve, and add f of 1 to it. If I take the area under
this curve, all the way down to here and then add
f of 1, what do I get? Upper bound on my sum. OK. So the sum i equals
1 to n f of i is upper bounded by that
guy, which is f of 1, plus my integral, which is
the area under the curve. The integral is the area under
that curve, starting here, and that contains
all these rectangles, and then I just add in f
of 1 to get an upper bound. Now, for the lower bound,
think about shifting the whole curve left by one. That goes to there,
this goes to here, that goes to here, that goes to
there, and that goes to there. The area under the
curve did not change when I shifted it left by one. This is now my area
under the curve. Stops here. What do I get when I take that
area and add in this last box, f of n? A lower bound,
because it’s contained in all the rectangles. Now, what’s really weird
about these formulas, do they look familiar? Yeah. Yeah. I switched them. Yeah. They’re the same formulas
we had over here, except we switched the direction
on the less than and greater than signs. Well, I swapped f of 1
and f of n, however you want to think about it. The lower bound
here in that case became the upper
bound in this case. Is that possible that the lower
bound became the upper bound? Yeah. Yeah, because what
really happened here– which is the big term
in this case, fn or f1? f1 is the big term
because it’s decreasing, so it’s totally symmetric. All right? The proof was very
similar, so the nice thing is you’ve only got to remember
the bounds are now simple for any sum as long as an
increasing or decreasing, it’s the same as the integral. The lower bound is the smaller
of the first and last term, and the upper bounds are larger
of the first and last term. Very easy to remember. Probably don’t even
need the crib sheet for it, although to be safe,
want to write that down. So now it’s easy to compute
good bounds on the sum of the inverse square roots. Any questions there
before I go do it? So let’s take the case
where we’re summing 1 over square root of i. So we compute the integral of
1 over square root of x dx. That equals the square root of x
over 1/2, evaluated at n and 1. That equals 2 square
root of n minus 1, or two square root of n minus
2, and now we can bound the sum. The upper bound is f
of 1 plus the integral. The lower bound is f
of n plus the integral. What is f of 1? One? One over the square
root of 1 is 1. What is f of n? One over the square root of n. Small. So these bounds are
pretty close here, right? In fact, this gets really
tiny as n gets big, so I’m just going
to replace this with 2 square root of n minus
2 and make it a strict lower bound, and this–
cancel there– I get 2 square root of n minus 1. Wow, these bounds are great. They’re within one for all n. That’s really good. So we can rewrite this in
terms of what really matters. What really matters
in these bounds? How fast is this
function growing? AUDIENCE: 2 root n. PROFESSOR: 2 root n. That’s what really matters,
so let’s write that down. So this says that the
sum i equals 1 to n of 1 over square root i
equals 2 square root of n– we have a minus delta n, where
delta is between 1 and 2. And so if I use the tilde
notation, what would I write down here for the tilde? Past the tilde? I don’t want to mess– AUDIENCE: Tilde. PROFESSOR: –I don’t
want to keep track of all the delta stuff as n gets big. AUDIENCE: 2 root n. PROFESSOR: 2 root
n, because this term over that goes to 0 as
n gets large, so let’s just check that. So we take the limit as n
goes to infinity of 2 root n minus delta n over 2 root n. I’m just checking
the definition now. That’s what the
definition would be. Equals 1 minus the
limit as n goes to infinity of 2 over 2 root n. This is 0. So it equals 1. And so now you know that the
sum of the first n inverse square roots grows as 2 root
n, which is the integral. Yeah. AUDIENCE: [INAUDIBLE] dropped
off the lower bound that f of n was 1 over root n? PROFESSOR: Yeah, I dropped it
off, because it was so tiny and going to zero, I just
made a strict less than. In fact, yes. I don’t hurt myself
by dropping it off. In fact, the lower
bound was a little bit– I made a little
weaker lower bound. So this is still true. I just– it wasn’t as
tight as it used to be, so I could keep it around. Yeah, it doesn’t hurt
to keep it around, then it’s a less
than or equal there. And now this would be
something like that. So I could keep it around,
but I’m going to get rid of it anyway, because I’m going
to go to the tilde notation, and as n gets big,
this is really tiny. So in this case, the
bounds are great. You can nail it
pretty much right on. Yeah. AUDIENCE: You said one
over n still there, the number is bigger than
it would normally be, so when you take it
out, it becomes smaller, so how could you go to
a less than [INAUDIBLE]? PROFESSOR: Well, you’re
saying you don’t like the fact I dropped it here? AUDIENCE: Yes. When you drop it, why do you
go to a less than instead of [INAUDIBLE]? PROFESSOR: Oh, because I’ve got
a bigger bound that I made less when I dropped it. I took something away, so I
know I could never equal this, because I know it’s
bigger than this. I know that the real answer
has to be at least this big, and so it has to be bigger
than something smaller. That’s why I did it. Any other questions? We’ll get more practice tomorrow
and next time with this stuff.

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