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## Alternating series test for convergence

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# Alternating series test

AP Calc: LIM‑7 (EU), LIM‑7.A (LO), LIM‑7.A.10 (EK)

## Video transcript

- [Narrator] Let's now expose ourselves to another test of convergence, and that's the Alternating Series Test. I'll explain the Alternating Series Test and I'll apply it to an
actual series while I do it to make the... Explanation of the Alternating Series Test a little bit more concrete. Let's say that I have some
series, some infinite series. Let's say it goes from N equals
K to infinity of A sub N. Let's say I can write it as or I can rewrite A sub N. So let's say A sub N, I can write. So A sub N is equal to
negative one to the N, times B sub N or A sub N is equal to
negative one to the N plus one times B sub N where B sub N is greater
than or equal to zero for all the Ns we care about. So for all of these integer
Ns greater than or equal to K. If all of these things, if
all of these things are true and we know two more things, and we know number one, the
limit as N approaches infinity of B sub N is equal to zero. Number two, B sub N is
a decreasing sequence. Decreasing... Decreasing sequence. Then that lets us know that
the original infinite series, the original infinite series, is going to converge. So this might seem a little
bit abstract right now. Let's actually show, let's
use this with an actual series to make it a little bit more, a little bit more concrete. Let's say that I had the series, let's say I had the series
from N equals one to infinity of negative one to the N over N. We could write it out
just to make this series a little bit more concrete. When N is equal to one, this
is gonna be negative one to the one power. Actually, let's just
make this a little bit, let's make this a little
bit more interesting. Let's make this negative
one to the N plus one. When N is equal to one, this is gonna be negative
one squared over one which is gonna be one. Then when N is two, it's
gonna be negative one to the third power which is gonna be negative one half. So it's minus one half plus one third minus one fourth plus minus and it keeps going
on and on and on forever. Now, can we rewrite
this A sub N like this. Well sure. The negative one to the N plus one is actually explicitly called out. We can rewrite our A sub N, so let me do that. So negative, so A sub N which
is equal to negative one to the N plus one over N. This is clearly the same
thing as negative one to the N plus one times one over N which is, which we can then say this thing right over
here could be our B sub N. This right over here is our B sub N. We can verify that our
B sub N is going to be greater than or equal to zero
for all the Ns we care about. So our B sub N is equal to one over N. Clearly this is gonna be
greater than or equal to zero for any, for any positive N. Now what's the limit? As B sub N, What's the limit of B sub
N as N approaches infinity? The limit of, let me
just write one over N, one over N, as N approaches
infinity is going to be equal to zero. So we satisfied the first constraint. Then this is clearly a decreasing sequence as N increases the denominators
are going to increase. With a larger denominator, you're going to have a lower value. We can also say one
over N is a decreasing, decreasing sequence for the Ns that we care about. So this satifies, this
is satisfied as well. Based on that, this thing is always, this thing right over here is always greater than or equal to zero. The limit, as one over
N or as our B sub N, as N approaches infinity,
is going to be zero. It's a decreasing sequence. Therefore we can say
that our originial series actually converges. So N equals 1 to infinity of negative one to the N plus over N. And that's kind of interesting. Because we've already seen that if all of these were positive, if all of these terms were positive, we just have the Harmonic Series, and that one didn't converge. But this one did, putting these
negatives here do the trick. Actually we can prove this
one over here converges using other techniques. Maybe if we have time, actually in particular
the limit comparison test. I'll just throw that out
there in case you are curious. So this is a pretty powerful tool. It looks a little bit about
like that Divergence Test, but remember the
Divergence Test is really, is only useful if you want
to show something diverges. If the limit of, if
the limit of your terms do not approach zero, then you say okay, that
thing is going to diverge. This thing is useful because you can actually
prove convergence. Once again, if something does not pass the alternating series test, that does not necessarily
mean that it diverges. It just means that you couldn't use the Alternating Series Test
to prove that it converges.

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