Originally Posted by

**arbolis** Just want to be sure about my results.

Say whether it converges or not :

1)$\displaystyle 1-\frac{1}{2}+\frac{2}{3}-\frac{1}{3}+\frac{2}{4}-\frac{1}{4}+\frac{2}{5}-\frac{1}{5}+...$. I wrote it as $\displaystyle \sum_{n=2}^{+\infty} \frac{2}{n}+\sum_{n=2}^{+\infty} -\frac{1}{n}=$ The harmonic series, so divergent.

2) $\displaystyle 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+...$ I wrote it as $\displaystyle \sum_{n=1}^{+\infty} \frac{(-1)^{n+1}}{2n-1}$. Taking the limit when $\displaystyle n$ tends to $\displaystyle +\infty$ of $\displaystyle \frac{1}{2n+1}$ and seeing it's equal to $\displaystyle 0$, the series converges.