I got the following test question:
Series $\sum_{n= 0}^{\infty}a_n$ converges. Which of the series below must also converge:
1) $\sum_{n= 0}^{\infty}na_n$
2) $\sum_{n= 0}^{\infty}a_n^2$
3 $\sum_{n= 0}^{\infty}(-1)^na_n$
To me it looks like non of the above should necessarily converge.
1) $\sum_{n= 1}^{\infty}\frac{1}{n^2}$ converges but $\sum_{n= 1}^{\infty}\frac{1}{n}$ does not.
2) $\sum_{n= 1}^{\infty}(-1)^n\frac{1}{\sqrt{n}}$ converges (Leibniz criterion) but $\sum_{n= 1}^{\infty}\frac{1}{n}$ does not.
3) $\sum_{n= 1}^{\infty}(-1)^n\frac{1}{n}$ converges but $\sum_{n= 1}^{\infty}(-1)^n(-1)^n\frac{1}{n}=\sum_{n= 1}^{\infty}\frac{1}{n}$ does not.
What am I missing?
