2.10.13 Number series

Problem: Make sure that the Leibniz test cannot be applied to the series \( \sum_{n=1}^{\infty} N_{n} \) with the terms \( (k \in N) \) indicated below. Examine this series for convergence in other ways. \[ a_{n}=\left\{\begin{array}{cc} \frac{1}{3^{k}}, & n=2 k-1 \\ -\frac{1}{2^{k}}, \quad n=2 k \end{array}\right. \]