[Math] Sum of $1-\frac{2^2}{5}+\frac{3^2}{5^2}-\frac{4^2}{5^3}+….$

Question

$1-\frac{2^2}{5}+\frac{3^2}{5^2}-\frac{4^2}{5^3}+….$

How can we find sum of above series upto infinite terms?

I don't know how to start and just need some hint.

Answer 1

Note that

$$\frac{1}{1 - x} = 1 + x + x^2 + x^3 + \dots$$ $$\frac{d}{dx}\left(\frac{1}{1 - x}\right) = 1 + 2x + 3x^2 + 4x^3 + \dots$$ $$x\cdot\frac{d}{dx}\left(\frac{1}{1 - x}\right) = x + 2x^2 + 3x^3 + 4x^4 + \dots$$ $$\frac{d}{dx}\left(x\cdot\frac{d}{dx}\left(\frac{1}{1 - x}\right)\right) = 1 + 2^2x + 3^2x^2 + 4^2x^3 + \dots$$

Now determine LHS and substitute in $x = -\frac{1}{5}$ (I used Wolfram):

$$-\frac{x + 1}{(x - 1)^3} = 1 + 2^2x + 3^2x^2 + 4^2x^3 + \dots$$ $$\frac{25}{54} = 1 - \frac{2^2}{5} + \frac{3^2}{5^2} - \frac{4^2}{5^3} + \dots$$