# Series, is this allowed?

1. Apr 3, 2013

### iRaid

1. The problem statement, all variables and given/known data
Well I have a series that I solved one way, but my professor solved another and I'm wondering if my way is ok.
$$\sum\limits_{n=1}^\infty \frac{(-5)^{2n}}{n^{2}9^{n}}$$

2. Relevant equations

3. The attempt at a solution
Alright well I started out by changing it to:
$$\sum\limits_{n=1}^\infty \frac{1}{n^{2}}(\frac{(-5)^{2n}}{9^{n}})=\sum\limits_{n=1}^\infty \frac{1}{n^{2}}(\frac{25}{9})^{n}$$

So I concluded since the second part is a geometric series with r>1, it's divergent.

Is this allowed?

2. Apr 3, 2013

### Infrared

Your manipulations are legal, but how do you conclude that just because the "second part" of the series is divergent, that the entire series is divergent. There is, after all, a factor of $\frac{1}{n^2}$ that reduces each term. You could look at long term behavior of the terms however and note that $\displaystyle\lim_{n\rightarrow \infty} {\frac{(\frac{25}{9})^n}{n^2}} = \infty$.

3. Apr 3, 2013

### iRaid

Very interesting, never thought about doing that. Could of just done the standard ratio test, which I think would work out easier, but hey this is a good way to do this problem I think.

Thanks.