proving series convergence by definition

Question

I have a question in the homework, and I tried to prove it
but I don't know if it is correct?

I want to prove that the series $$ \sum_{n=1}^{\infty} \frac{z^n}{n(n+1)} $$ is absolutely convergent for $|z|\le 1$.

My answer is by Abel's Theorem:

$$\left|\frac{z^n}{n(n+1)} \right| \lt \frac{1}{n^2}$$ which is convergent series , thus our series is absolutely convergent
Is This True ??

Answer 1

Hint: Use the comparison test. $$\left|\dfrac{z^n}{n(n+1)}\right|\leq\dfrac{1}{n^2}$$ Use also the fact that the sum $$\sum^\infty_{n=1}\dfrac{1}{n^2}=\dfrac{\pi^2}{6}$$ To simply prove that $\sum^\infty_{n=1}\dfrac{1}{n^2}$ converges, use the integral test.

Answer 2

Follows just by comparison to $\frac{1}{n^{2}}$ as you showed; Abel's theorem is not needed.