- [Voiceover] Let's say that we have an infinite series S
so that's the sum from n = 1
to infinity of a sub n.
We could write it out a sub 1 plus a sub 2
and we're just going to go on and on and on for infinity.
We're going to go on and on and on forever.
So, let's say, and I've written it in very general terms
let's say we have a formula for the partial sums of S.
We know that S sub n is equal
to 2n to the third
over n plus 1
times n plus 2.
Now, my question to you is, based on what I've just told you
S is the sum in a very general way written
this infinite series but I have the partial sum.
The sum of the first n terms of S is given
by this formula right over here does
this series converge or diverge?
Does this thing converge to some finite value
or is it unbounded and does it diverge?
Well, one way to think about this is
the idea that our infinite series S
is just the limit as n approaches
infinity of our partial sums.
So, what do we mean by that?
Well, you could a sequence of partial sums here.
You have S sub 1, S sub 2, S sub 3
and you keep going so this
would be the sum of the first term.
This would be the sum of the first 2 terms.
This would be the sum of the first 3 terms
and just think about what happens to this sequence
as n right over here
approaches infinity
because that's what this series is.
It's the sum of the first, I guess
you could say the first, infinite terms.
It's the sum of all, you have
an infinite number of terms here.
Well, let's think about what this.
The limit is n approaches infinity of S sub n.
That's just going to be
the limit as n approaches
infinity of this business right over here.
2n to the third power over n plus 1
times n plus 2 and there's
several ways you could evaluate this.
One way is you could just realize,
"Hey, look in the bottom this is going
"to be a second degree polynomial."
On up here, you have a third degree
so the numerator is gonna grow faster
than the denominator so this is going to be unbounded.
So that will immediately tell you
well this is gonna approach infinity
so S is going to
diverge
but if you wanna do it a little bit
less hand wavy than that we can
actually do a little bit more algebra.
Limit as n approaches infinity, 2n
to the third power
over, let's multiply this out,
n squared plus 3n
plus 2
and, let's see, we can divide the numerator
and the denominator by n squared.
So, this is going to be the limit
as n approaches infinity of, if we divide
the numerator by n squared, you're going
to have, actually, let's divide the numerator
and, well yeah let's divide it by n squared
so, if we divide the numerator by n squared,
we're gonna have 2n and then the denominator
divided by n squared you're gonna have
1 plus 3 over n
plus 2 over n squared.
Now, when you look at it like this,
it becomes pretty clear this thing
as n approaches infinity, this thing is gonna
towards infinity but this thing down here
the denominator this is gonna go towards 0.
This is gonna go towards 0
so the denominator's gonna go towards 1.
So, this whole thing, is the limit
is gonna go to infinity and since the limit
of the partial sums goes to infinity
that mean that this infinite series
is not going to be a finite value.
It's just going to diverge.
So, this character right over here
is going
to diverge.
In order for it to have converged,
this thing should have come, this limit
should have been some finite value.
So, hopefully, that makes sense.
All we say is, "Look, infinite series,
"we had a formula for the partial sum
"of the first n terms and then we said oh look
"the series itself, the infinite series,
"you could view it as a limit of,
"as n approaches infinity, of the partial sum
"S sub n and we said hey, that approach infinity
"this thing is diverging."