1. ## Infinite Series

Determine convergence or divergence:

$\displaystyle a_n = \frac {1+(-1)^n}{n^2}$

Does the this function converge and is the limit equal to 0? That is what I found it to be.

2. Originally Posted by saiyanmx89
Determine convergence or divergence:

$\displaystyle a_n = \frac {1+(-1)^n}{n^2}$

Does the this function converge and is the limit equal to 0? That is what I found it to be.
yes. one way to see this is to use the squeeze theorem

3. hi

I guess you could do

$\displaystyle \frac{1 +(-1)^{n}}{n^{2}} \, \leq \, \frac{(-1)^{n}}{n^{2}}$

And the series to the right is converging by Leibniz convergence theorem, because the series is alternating and decreasing.

4. Originally Posted by Twig
hi

I guess you could do

$\displaystyle \frac{1 +(-1)^{n}}{n^{2}} \, \leq \, \frac{(-1)^{n}}{n^{2}}$

[snip]
This inequality is false for even values of n.

5. $\displaystyle (-1)^n$ is bounded and $\displaystyle \frac1{n^2}\to0$ as $\displaystyle n\to\infty,$ thus $\displaystyle \frac{(-1)^n}{n^2}\to0$ as $\displaystyle n\to\infty,$ and the conclusion follows.