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Sum of an infinite series of (a quadratic in n) over a factorial, in terms of e

The same question, asked 2 times: 2021 Paper 26*, 1 Feb 2023. Every wording below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2021 Paper 26

    62
    The sum of the series ∑n=1∞n2+6n+10(2n+1)!\sum_{n= 1}^{\infty} \frac{n^{2}+ 6n+ 10}{(2n+ 1)!} is equal to:

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  2. Question 2

    1 Feb 2023

    152
    The sum ∑n=1∞2n2+3n+4(2n)!\sum_{n = 1}^{\infty} \frac{2n^{2}+ 3n + 4}{(2n)!} is equal to:

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