2.10.130 Number series

Condition: Using the definition of convergence of a series, prove the convergence of the series and find its sum. a) \( \sum_{n=0}^{\infty} \frac{1}{(n+7)(n+9)} \); b) \( \sum_{n=1}^{\infty} \frac{5^{n+1}-7 \cdot 3^{n}}{10^{n}} \).