2.10.133 Number series

condition: a) Prove that the series converges, b) Find the sum of the series accurate to 0.00001: 1) \( \sum_{n=1}^{\infty} \frac{n^{2} \cdot 2^{n}}{(n+2) !} ; \quad \sum_{n=1}^{\infty} \frac{(-1)^{n}}{\sin ^{2} n+n} \).