Prove or disprove: $$\sum_{k=0}^{\infty}(-1)^{k}\frac{1}{2k+1}$$ is convergent.
I have started trying with ratio test but at the end I had $1$ as result which is bad :D
(Would you even recommend me trying ratio test on an alernating series?)
Then I have used another theorem but I don't know its name. Show that it's a zero-sequence and then show it's monotonic decreasing:
$$\lim_{k\rightarrow\infty}\frac{1}{2k+1}=0$$
Now show it's monotonic decreasing:
$$\frac{1}{2k+1}\geq \frac{1}{2(k+1)+1}$$
$$\frac{1}{2k+1} \geq \frac{1}{2k+3}$$
$$1 \geq \frac{2k+1}{2k+3}$$
Thus it's monotonic decreasing. So all in all, the series converges.
Did I do it correctly?