Partial fraction decomposition to find sum of telescoping series

So what we're going
to attempt to do

is evaluate this
sum right over here,

evaluate what this
series is, negative 2

over n plus 1 times n plus
2, starting at n equals 2,

all the way to infinity.

And if we wanted to see what
this looks like, it starts at n

equals 2.

So when n equals 2, this
is negative 2 over 2

plus 1, which is 3, times
2 plus 2, which is 4.

Then when n is equal to 3,
this is negative 2 over 3

plus 1, which is 4, times
3 plus 2, which is 5.

And it just keeps going
like that, negative 2

over 5 times 6.

And it just keeps
going on and on and on.

And now, it looks pretty clear
that each successive term

is getting smaller.

And it's getting
smaller reasonably fast.

So it's reasonable to assume
that even though you have

an infinite number
of terms, it actually

might give you a finite value.

But it doesn't jump out
at me, at least the way

that I've looked
at it right now,

as to what this sum
would be, or how

to actually figure out that sum.

So what I want you to do
now is pause this video.

And I'm going to give you a hint
about how to think about this.

Try to dig up your memories
of partial fraction expansion,

or partial fraction
decomposition,

to turn this expression into
the sum of two fractions.

And that might help us think
about what this sum is.

So I'm assuming you've
given a go at it.

So let's try to
manipulate this thing.

And let's see if we can rewrite
this as a sum of two fractions.

So this is negative
2 over-- and I'm

going to do this in two
different colors-- n plus 1

times n plus 2.

And we remember from our
partial fraction expansion

that we can rewrite this as
the sum of two fractions,

as A over n plus 1
plus B over n plus 2.

And why is this reasonable?

Well, if you're
adding two fractions,

you want to find a
common denominator, which

would be a multiple of
the two denominators.

This is clearly a multiple of
both of these denominators.

And we learned in partial
fraction expansion

that whatever we have up here,
especially because the degree

here is lower than the degree
down here, whatever we have

up here is going to be a degree
lower than what we have here.

So this is a first-degree
term in terms of n,

so these are going to be
constant terms up here.

So let's figure out
what A and B are.

So if we perform
the addition-- well,

let's just rewrite
both of these with

the same common denominator.

So let's rewrite
A over n plus 1,

but let's multiply the numerator
and denominator by n plus 2.

So we multiply the numerator
by n plus 2 and the denominator

by n plus 2.

I haven't changed the value
of this first fraction.

Similarly, let's do the same
thing with B over n plus 2.

Multiply the numerator and the
denominator by n plus 1, so

n plus 1 over n plus 1.

Once again, I haven't change
the value of this fraction.

But by doing this, I now
have a common denominator,

and I can add.

So this is going to be equal
to n plus 1 times n plus 2 is

our denominator.

And then our numerator--
let me expand it out.

This is going to be,
if I distribute the A,

it is An plus 2A.

So let me write
that, An plus 2A.

And then let's distribute
this B, plus Bn plus B.

Now, what I want to do
is I want to rewrite this

so I have all of the n terms.

So for example, An plus Bn--
I could factor an n out.

And I could rewrite that
as A plus B times n, those

two terms right over there.

And then these two
terms, the 2A plus B,

I could just write it
like this, plus 2A plus B.

And, of course, all of that is
over n plus 1 times n plus 2.

So how do we solve for A and B?

Well, the realization
is that this thing

must be equal to negative 2.

These two things must
be equal to each other.

Remember, we're making
the claim that this,

which is the same thing
as this, is equal to this.

That's the whole reason
why we started doing this.

So we're making the
claim that these two

things are equivalent.

We're making this claim.

So everything in the numerator
must be equal to negative 2.

So how do we do that?

It looks like we have
two unknowns here.

To figure out two unknowns, we
normally need two equations.

Well, the realization
here is, look,

we have an n term on
the left-hand side here.

We have no n term here.

So you literally could view
this, instead of just viewing

this as negative
2, you could view

this as negative 2 plus
0n, plus 0 times n.

That's not "on."

That's 0-- let me write
it this way-- 0 times n.

So when you look at
it this way, it's

clear that A plus B is
the coefficient on n.

That must be equal to 0.

A plus B must be equal to 0.

And this is kind of
bread-and-butter partial

fraction expansion.

We have other videos on that
if you need to review that.

And the constant part, 2A plus
B, is equal to negative 2.

And so now we have two
equations in two unknowns.

And we could solve it a
bunch of different ways.

But one interesting way is
let's multiply the top equation

by negative 1.

So then this becomes negative
A minus B is equal to-- well,

negative 1 times 0 is still 0.

Now we can add these
two things together.

And we are left with 2A minus
A is A, plus B minus B-- well,

those cancel out.

A is equal to negative 2.

And if A is equal to
negative 2, A plus B is 0,

B must be equal to 2.

Negative 2 plus 2 is equal to 0.

We solved for A. And then I
substituted it back up here.

So now we can rewrite all
of this right over here.

We can rewrite it as
the sum-- and actually,

let me do a little bit instead.

Let me just write it as
a finite sum as opposed

to an infinite sum.

And then we can just take the
limit as we go to infinity.

So let me rewrite it like this.

So this is the sum from n
equals 2-- instead to infinity,

I'll just say to capital
N. And then later, we

could take the limit as this
goes to infinity of-- well,

instead of writing this, I can
write this right over here.

So A is negative 2.

So it's negative
2 over n plus 1.

And then B is 2,
plus B over n plus 2.

So once again, I've just
expressed this as a finite sum.

Later, we can take the limit
as capital N approaches

infinity to figure out
what this thing is.

Oh, sorry, and B-- let
me not write B anymore.

We now know that B
is 2 over n plus 2.

Now, how does this actually
go about helping us?

Well, let's do what
we did up here.

Let's actually write out what
this is going to be equal to.

This is going to be
equal to-- when n is 2,

this is negative 2/3, so
it's negative 2/3, plus 2/4.

So that's n equals-- let me
do it down here, because I'm

about to run out of real estate.

That is when n is equal to 2.

Now, what about when
n is equal to 3?

When n is equal to 3, this
is going to be negative 2/4

plus 2/5.

What about when n is equal to 4?

I think you might see a pattern
that's starting to form.

Let's do one more.

When n is equal to
4, well, then, this

is going to be
negative 2/5-- let

me do that same blue color--
negative 2/5 plus 2/6.

And we're just
going to keep going.

Let me scroll down to
get some space-- we're

going to keep going all the
way until the N-th term.

So plus dot dot dot plus
our capital N-th term,

which is going to be negative
2 over capital N plus 1 plus 2

over capital N plus 2.

So I think you might
see the pattern here.

Notice, from our first when
n equals 2, we got the 2/4.

But then when n equals 3,
you had the negative 2/4.

That cancels with that.

When n equals 3, you had 2/5.

Then that cancels when n
equals 4 with the negative 2/5.

So the second term cancels
with-- the second part,

I guess, for each
n, for each index,

cancels out with the first
part for the next index.

And so that's just going to
keep happening all the way

until n is equal to capital N.

And so this is going to
cancel out with the one right

before it.

And all we're going
to be left with

is this term and this
term right over here.

So let's rewrite that.

So we get-- let's
get more space here.

This thing can be rewritten
as the sum from lowercase n

equals 2 to capital
N of negative 2

over n plus 1 plus
2 over n plus 2

is equal to-- well, everything
else in the middle canceled

out.

We're just left
with negative 2/3

plus 2 over capital N plus 2.

So this was a huge
simplification right over here.

And remember, our original sum
that we wanted to calculate,

that just has a limit as
capital N goes to infinity.

So let's just take the limit
as capital N goes to infinity.

So let me write it this way.

Well, actually, let
me write it this way.

The limit-- so we can
write it this way.

The limit as
capital N approaches

infinity is going to be equal
to the limit as capital N

approaches infinity
of-- well, we just

figured out what this is.

This is negative 2/3 plus
2 over capital N plus 2.

Well, as n goes to
infinity, this negative 2/3

doesn't get impacted at all.

This term right over here, 2
over an ever larger number,

over an infinitely
large number-- well,

that's going to go to 0.

And we're going to be
left with negative 2/3.

And we're done.

We were able to figure out the
sum of this infinite series.

So this thing right over here
is equal to negative 2/3.

And this type of series is
called a telescoping series--

telescoping, I should say.

This is a telescoping series.

And a telescoping series
is a general term.

So if you were to
take its partial sums,

it has this pattern right over
here, where, in each term,

you're starting to
cancel things out.

So what you're left with
is just a fixed number

of terms at the end.

But either way,
this was a pretty--

it's a little bit hairy, but
it was a pretty satisfying

problem.