How to determine whether the series converges or diverges $$\begin{align*}
\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)}
\end{align*}$$

I apprpach this problem by using the ratio test.

$$ \lim_{k\rightarrow \infty}\frac{\dfrac{(-1)^{k+1}}{2(k+1)+1}}{\dfrac{(-1)^k}{(2k+1)}}$$

which after simplifying gets me $(2k)/(2k+3) + (1)/(2k+3)$

How does one simplify this to $\sqrt{k+1}$ as the answer it is very perplexing?
Hint. The alternating series $\displaystyle \sum_{k=0}^{\infty}(-1)^{k}a_k$ converges provided:

1) $0&lt; a_{k+1} \le a_k$ for $k$ greater than some index $N$.

2) $\lim_{k\to \infty}a_k = 0$

Just take $a_k:=\frac{1}{2k+1}$ here.
you can use the Alternating Series Test,

http://en.wikipedia.org/wiki/Alternating_series_test