Let's consider the alternating sign series...

$\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{\sqrt{n}}$ (1)

If we indicate with $\displaystyle a_{n}$ the general term of (1), you can verify that...

1) $\displaystyle \{|a_{n}|\}$ is a decreasing sequence

2) $\displaystyle \lim_{n \rightarrow \infty} a_{n}=0$

... so that (1) converges. Now, indicating with $\displaystyle \sigma$ the sum of (1), the questions I intend to indicate to your attention are...

a) is $\displaystyle \sigma$ rational or irrational?...

b) if $\displaystyle \sigma$ is irrational, can it be expressed as elementary function of rational numbers or other constants like $\displaystyle \pi$, $\displaystyle e$ and so one?...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$