We were being taught this subject today, however, the professor did not get to finish the lecture and mentioned something about testing the end points of the interval with another test of convergence, so they function doesn't diverge at the end points? How would I do this, as in checking the end points properly, just want to get a step ahead,

I was given the following example I reworked myself till I obtained the interval of convergence

$\displaystyle \sum_{n=1}^{\infty}\frac{(2x+3)^n}{n^2}$

I implemented a ratio test, here the end result

$\displaystyle |2x-3|\lim_{n\to\infty}(\frac{n}{n+1})^2$

$\displaystyle |2x-3|\lim_{n\to\infty}(1-\frac{1}{n+1})^2 = 1$

$\displaystyle = |2x+3|$

$\displaystyle -1\le2x+3\le1$

$\displaystyle -2\le x \le -1$

2. Here is a hint: the series at $\displaystyle x=-2$ and $\displaystyle x=-1$ are

$\displaystyle \sum_{n=1}^{\infty}\frac{(-1)^n}{n^2}=-1+\frac{1}{2^2}-\frac{1}{3^2}+\frac{1}{4^2}-\cdots$

and

$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^2}=1+\frac{1}{2^2}+\ frac{1}{3^2}+\frac{1}{4^2}+\cdots.$