2.10.7 Number series

Problem: Examine the series for absolute and conditional convergence: \[ \sum_{n=1}^{\infty}(-1)^{n+1} \cdot a_{n}, \quad \text { where } a_{n}=\frac{1 \cdot 4 \cdot \ldots \cdot(3 n-2)}{3 \cdot 5 \cdot \ldots \cdot(2 n+1)} \]