5
votes
1answer
126 views

Interesting phenomenon with the $\zeta(3)$ series

I noticed that if one takes certain partial sums of the series for $\zeta(3)$: $$\zeta(3) = \sum_{n=1}^{\infty} \frac{1}{n^3} \approx \sum_{n=1}^{N} \frac{1}{n^3}$$ an interesting phenomenon occurs ...
2
votes
2answers
283 views

Convergence of the series $\sum \frac{\sqrt{n+1}-\sqrt{n}}{n^x}$

Could you help me to understand for which $x$ this series converge $\displaystyle\sum \frac{\sqrt{n+1}-\sqrt{n}}{n^x}$?