Question related to convergence of series $\sum_{n=1}^\infty (-1)^n\frac {x^2+n} {n^2}$.

Question

We have the series $\displaystyle\sum_{n=1}^\infty (-1)^n\frac {x^2+n} {n^2}$. Which test ensure that the series convergence for all real value of $x$ and how can we confirm that this series does not converge absolutely for any real vale of $x$.

MY TRY:I just used ratio test but I did not get any clue for the purpose.Thank you

Answer 1

For absolute convergence, since Nameless has taken care of standard convergence, note that $|(-1)^n\frac{x^2 + n}{n^2}| = \frac{x^2}{n^2} + \frac{1}{n}$. Does this sound any alarms for you? It should!

Answer 2

Check whether

(1) $\lim_n \frac{x^2 + n}{n^2} = 0$

(2)$ \left ( \frac{x^2 + n}{n^2} \right )_{n \geq 1}$ is a decreasing sequence.

Use this test

I'll give you some hints

(1) Just treat $x^2$ as a constant.

(2) Plot the function and once again fix $x^2$

yes it converges