The given series is,
$$\sum\limits_{n\geq 1}(\sqrt{n+1}-\sqrt{n})^{\alpha}(\ln(1+1/n))^{\beta}$$
With $\alpha, \beta \in \mathbb{R}$.
I don't know how to begin, noreven which criterion use to find the values of $\alpha$ and $\beta$ for which this series converges.
Best Answer
If you multiply the first term by $\frac{\sqrt{n+1}+\sqrt{n}}{\sqrt{n+1}+\sqrt{n}}$ and use Maclauring series expansion for the second term, you can compare the sum to $$ \frac{1}{2^{\alpha}}\sum_{k=1}^{\infty}\frac{1}{n^{\frac{\alpha}{2}+\beta}} $$ which converges for $\frac{\alpha}{2}{+\beta}>1$