Question

sum to infinity the series sigma n=1 to infinity (n+1)(2n+1)/(n+2)!

Answer 1

http://www.wolframalpha.com/input/?i=Sum%5B%28n%2B1%29%282n%2B1%29%2F%28n%2B2%29%21%2C+%7Bn%2C+1%2C+Infinity%7D%5D

Answer 2

\[\sum_{n=1}^\infty \frac{(n+1)(2n + 4 - 3)}{(n+2)!} = \sum_{n=1}^\infty \frac{2(n+1)(n + 2) - 3(n+1))}{(n+2)!} \\
= 2 \sum_{n=1}^\infty \frac 1 {n!} - 3\sum_{n = 1}^\infty \frac{n+2 - 1}{(n+2)!} = 2(e-1) - 3\left( \sum_{n=1}^\infty \frac{1}{(n+1)!} - \frac{1}{(n+2)!} \right ) \\= 2e - \frac 3 2 - 2 = 2e - \frac 7 2 \]

Answer 3

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