Solved: "What is∑n=0∞2n7+n6+n5+2n2n!"

Willie Kelley

Answered question

2022-04-19

What is

\sum_{n = 0}^{\infty} \frac{2 n^{7} + n^{6} + n^{5} + 2 n^{2}}{n!}

ze2m1ingkdvu

Beginner2022-04-20Added 16 answers

You have that if:

A (z) = \sum_{n \geq 0} a_{n} z^{n}

then:

z \frac{d}{d z} A (z) = \sum_{n \geq 0} n a_{n} z^{n}

In general, using D for the differentiation operator, and p a polyomial:

\sum_{n \geq 0} p (n) a_{n} z^{n} = p (z D) A (z)

So you are looking for:

(2 {(z D)}^{7} + {(z D)}^{6} + {(z D)}^{5} + 2 {(z D)}^{2}) e^{z}

at z=1

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