Get the solution to "I want to evaluate the sum∑n=2∞n4+3n2+10n+102n(n4+4)"

Cian Orr

Answered question

2022-02-24

I want to evaluate the sum

\sum_{n = 2}^{\infty} \frac{n^{4} + 3 n^{2} + 10 n + 10}{2^{n} (n^{4} + 4)}

Chettaf04

Beginner2022-02-25Added 7 answers

So once you have obtained

\frac{1}{2^{n}} (\frac{- 1}{n^{2} + 2 n + 2} + \frac{4}{n^{2} - 2 n + 2} + 1)

then observe the following:

n^{2} + 2 n + 2 = {(n + 1)}^{2} + 1

and

n^{2} - 2 n + 2 = {(n - 1)}^{2} + 1

. Which means the given sum becomes

\sum_{n = 2}^{\infty} [\frac{- 1}{2^{n} [{(n + 1)}^{2} + 1]} + \frac{1}{2^{n - 2} [{(n - 1)}^{2} + 1]}] + \sum_{n = 2}^{\infty} \frac{1}{2^{n}} = \sum_{n = 2}^{\infty} [u_{n} - u_{n + 2}] + \frac{1}{2}

where

u_{n} = \frac{1}{2^{n - 2} [{(n - 1)}^{2} + 1]}

If you calculate

\sum_{n = 2}^{\infty} [u_{n} - u_{n + 2}] = u_{2} + u_{3} = \frac{1}{2} + \frac{1}{10}

Hence the final sum is

2 \times \frac{1}{2} \times \frac{1}{10} = \dots

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