2.10.12 Number series

Problem: Make sure that Leibniz test can't be applied to the series \( \sum_{n=1}^{\infty} N_{n} \) with the terms given below \( (k \in N) \). Examine this series for convergence in other ways. \[ a_{n}=\left\{\begin{array}{lc} \frac{1}{\sqrt{k+1}+1}, \quad n=2 k-1 \\ -\frac{1}{\sqrt{k+1}-1}, \quad n=2 k \end{array}\right. \]