How to find the value of this infinite sum rigorously? the problem is about this sum:
$$\sum_{n=0}^{\infty}(-1)^n(\sqrt{n+1}-\sqrt{n})
$$
This is a convergent series, converger to a value around 0.76. My approach is quite simple:
$$\sum_{n=0}^{\infty}(-1)^n(\sqrt{n+1}-\sqrt{n})=2\sum_{k=1}^{\infty}(-1)^{k+1}\sqrt{k}
=2\left(\sum_{k=1}^{\infty}\sqrt{k}-2\sum_{k=1}^{\infty}\sqrt{2k}\right)\\=2(1-2\sqrt{2})\cdot\zeta(-1/2)
$$
The actural value of answer seems right checking by Wolfram Alpha, but series that does not converge appears in my approach. Is there a more rigorous way to get the answer?

Compute the partial sums
$$S_p=\sum_{n=0}^{p}(-1)^n \sqrt{n+1}-\sum_{n=0}^{p}(-1)^n\sqrt{n} $$ in terms of the generalized Riemann zeta function to obtain
$$S_p=2 \left(1-2 \sqrt{2}\right) \zeta \left(-\frac{1}{2}\right)+$$ $$(-1)^{p+1} \sqrt{2}\Bigg[\zeta \left(-\frac{1}{2},\frac{p+1}{2}\right)-2 \zeta
   \left(-\frac{1}{2},\frac{p+2}{2}\right)+\zeta
   \left(-\frac{1}{2},\frac{p+3}{2}\right)\Bigg]$$
The term  in brackets is quite small : $\sim -0.169557$ for $p=0$, $\sim -0.017590$ for $p=100$
You could be interested by the third part of this paper.
