Finding convergence zone/range for $\sum_{i=1}^\infty \frac{x^{n^2}}{n(n+1)}$ $$\sum_{i=1}^\infty \frac{x^{n^2}}{n(n+1)}$$
I used the ratio test and I end up with:
$$|x|*\frac{n}{n+2}$$
What steps do I need to take to continue?
Looking for hints or steps, not full solution/

Did you notice the powers of $\;x\;$ are not consecutive? They are $\;1,4,9,16,...\;$ , so the actual coefficients are
$$a_n:=\begin{cases}\cfrac1{\sqrt{n(n+1)}},\,&n\;\text{is a square}\\{}\\0,\,&\text{otherwise}\end{cases}$$
and thus
$$\lim\sup_{n\to\infty}\sqrt[n]{|a_n|}=\lim_{n\to\infty}\frac1{\sqrt[n]{\sqrt{n(n+1)}}}=1$$
and thus the convergence radius is $\;1\;$ .
