Resolved: "Calculate sum of an infinite series∑n=1∞(−1)n−1n2xn"

Neveah Stewart

Answered question

2022-03-19

Calculate sum of an infinite series

\sum_{n = 1}^{\infty} {(- 1)}^{n - 1} n^{2} x^{n}

zonadeenfoqueun6

Beginner2022-03-20Added 3 answers

Note that

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

and hence the given series

\sum_{n \geq 1} n^{2} x^{n} =: f (n; x)

can be written as

f (n; x) = \sum_{n \geq 1} n (n - 1) x^{n} + \sum_{n \geq 1} n x^{n} = f_{1} (n; x)

respectively. Let us determine these

f_{1} (n; x)

and

f_{2} (n; x)

separtely.
Now

f_{1} (n; x) = x^{2} \sum_{n \geq 1} n (n - 1) x^{n - 2} = x^{2} \frac{2}{{(1 + x)}^{3}} = \frac{2 x^{2}}{{(1 + x)}^{3}}

by using the fact

{(1 + x)}^{- 1} = \sum_{n \geq 1} x^{n}

and two times successive differentiation.
On the other hand

f_{2} (n; x) = - x \frac{1}{{(1 + x)}^{2}} = \frac{- x}{{(1 + x)}^{2}}

Therefore

f (n; x) = \frac{2 x^{2}}{{(1 + x)}^{3}} - \frac{x}{{(1 + x)}^{2}}

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