Is the series $\sum _{n=1}^{\infty } (-1)^n / {n^2}$ convergent or absolutely convergent? Is this series convergent or absolutely convergent? $$\sum _{n=1}^{\infty }\:(-1)^n \frac {1} {n^2}$$
Attempt:
I got this using Ratio Test: 
$$\lim_{n \to \infty} \frac{n^2}{(n+1)^2}$$

The root and ratio test, are testing of behavior similar to that of a geometric series.
So for example $\sum \frac{n}{2^n}$ although not geometric is similar to a geometric and here root and ration will give an answer.
If the series does not behave like a geometric series the root and ratio are useless,
second kind of series are those whose behavior is similar to a harmonic, like your example $\sum\frac{1}{n^2}$ here the integral, majorization/minorization, or the ratio comparison test will be successful.  
