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## Integral Calculus (2017 edition)

### Unit 10: Lesson 2

Infinite sequences# Worked example: sequence convergence/divergence

AP.CALC:

LIM‑7 (EU)

, LIM‑7.A (LO)

, LIM‑7.A.1 (EK)

, LIM‑7.A.2 (EK)

How can we tell if a sequence converges or diverges? See Sal in action, determining the convergence/divergence of several sequences. Created by Sal Khan.

## Video transcript

So we've explicitly defined
four different sequences here. And what I want
you to think about is whether these sequences
converge or diverge. And remember,
converge just means, as n gets larger and
larger and larger, that the value of our sequence
is approaching some value. And diverge means that it's
not approaching some value. So let's look at this. And I encourage you
to pause this video and try this on your own
before I'm about to explain it. So let's look at this first
sequence right over here. So the numerator n plus 8 times
n plus 1, the denominator n times n minus 10. So one way to think about
what's happening as n gets larger and larger is look
at the degree of the numerator and the degree of
the denominator. And we care about the degree
because we want to see, look, is the numerator growing
faster than the denominator? In which case this thing
is going to go to infinity and this thing's
going to diverge. Or is maybe the denominator
growing faster, in which case this might converge to 0? Or maybe they're growing
at the same level, and maybe it'll converge
to a different number. So let's multiply out the
numerator and the denominator and figure that out. So n times n is n squared. n times 1 is 1n, plus 8n is 9n. And then 8 times 1 is 8. So the numerator is n
squared plus 9n plus 8. The denominator is
n squared minus 10n. And one way to
think about it is n gets really, really, really,
really, really large, what dominates in the
numerator-- this term is going to represent most of the value. And this term is going to
represent most of the value, as well. These other terms
aren't going to grow. Obviously, this 8
doesn't grow at all. But the n terms aren't going
to grow anywhere near as fast as the n squared terms,
especially for large n's. So for very, very
large n's, this is really going
to be approaching n squared over n squared, or 1. So it's reasonable to
say that this converges. So this one converges. And once again, I'm not
vigorously proving it here. Or I should say
I'm not rigorously proving it over here. But the giveaway is that
we have the same degree in the numerator
and the denominator. So now let's look at
this one right over here. So here in the numerator
I have e to the n power. And here I have e times n. So this grows much faster. I mean, this is
e to the n power. Imagine if when you
have this as 100, e to the 100th power is a
ginormous number. e times 100-- that's just 100e. Grows much faster than
this right over here. So this thing is just
going to balloon. This is going to go to infinity. So we could say this diverges. Now let's look at this
one right over here. Well, we have a
higher degree term. We have a higher
degree in the numerator than we have in the denominator. n squared, obviously, is going
to grow much faster than n. So for the same reason
as the b sub n sequence, this thing is going to diverge. The numerator is going
to grow much faster than the denominator. Or another way to think
about it, the limit as n approaches infinity
is going to be infinity. This thing's going
to go to infinity. Now let's think about
this right over here. So as we increase
n-- so we could even think about what the
sequence looks like. When n is 0, negative
1 to the 0 is 1. When n is 1, it's
going to be negative 1. When n is 2, it's going to be 1. And so this thing is
just going to keep oscillating between
negative 1 and 1. So it's not unbounded. It's not going to go to
infinity or negative infinity or something like that. But it just oscillates
between these two values. So it doesn't converge
to one particular value. So even though this one
isn't unbounded-- it doesn't go to infinity-- this
one still diverges. It doesn't go to one value. So let me write that down. This one diverges.