Question

Find the sum of the infinite series $1^2-\frac{2^2}{5}+\frac{3^2}{5^2}-\frac{4^2}{5^3}+\frac{5^2}{5^4}-\frac{6^2}{5^5}+....$

Answer 1

Assuming $x=\frac{1}{5}$, we have

$S=1-4x+9x^2-16x^3+25x^2-36x^5+\dots \dots$

$\Rightarrow 3xS=3x-12x^2+27x^3-48x^4+75x^5-\dots$

$\Rightarrow 3x^{2}S=3x^2-12x^3+27x^4-48x^5\dots$

$\Rightarrow x^{3}S=x^3-4x^4+9x^5\dots$

Adding the above series, we get

$(1+3x+3x^2+x^3)S=1-x$

$\Rightarrow (1+x{)}^{3}S=1-x$

$\Rightarrow S=\frac{1-x}{(1+x{)}^3}=\frac{25}{54}[\because x=\frac{1}{5}]$