let-q-R-q-lt-1-show-that-k-0-k-2-q-k-q-2-q-1-q-3-

Question Number 130921 by pticantor last updated on 30/Jan/21

l e t q \in R / ∣ q ∣< 1

s h o w t h a t

\underset{k = 0}{\sum^{+ \infty}} k^{2} q^{k} = \frac{q^{2} + q}{{(1 - q)}^{3}}

Answered by Dwaipayan Shikari last updated on 30/Jan/21

S = \underset{k = 0}{\sum^{\infty}} k^{2} q^{k} = 1^{2} q + 2^{2} q^{2} + 3^{2} q^{3} + \dots .

S (1 - q) = q + (2^{2} - 1^{2}) q^{2} + (3^{2} - 2^{2}) q^{3} + \dots

S {(1 - q)}^{2} = q + 2 q^{2} + 2 q^{3} + 2 q^{4} + \dots

S {(1 - q)}^{2} = \frac{2 q}{(1 - q)} - q \Rightarrow S = \frac{q^{2} + q}{{(1 - q)}^{3}}