# 2017-13 Infinite series with recurrence relation

Let $$a_0 = a_1 =1$$ and $$a_n = n a_{n-1} + (n-1) a_{n-2}$$ for $$n \geq 2$$. Find the value of
$\sum_{n=0}^{\infty} (-1)^n \frac{n!}{a_n a_{n+1}}.$

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Let $$A$$ and $$B$$ be $$n\times n$$ matrices. Prove that if $$n$$ is odd and both $$A+A^T$$ and $$B+B^T$$ are invertible, then $$AB\neq 0$$.