Question

$1+2x+3x^2+4x^3+.....\infty ;|x|<1$.

Answer 1

Let $S=1+2x+3x^2+3x^2+4x^3...\infty \left(1\right)$

Multiply the above equation by x

$Sx=x+2x^2+3x^3+4x^4...\infty \left(2\right)$

subtract equation (2)form equation(1)

$S-Sx=1+x+x^2+x^3+x^4+...$

$S\left(1.x\right)=1+x+x^2+x^3+x^4+...\left(3\right)$

equation (3)forms a G.p with first term is 1, and common difference x sum of infinite terms of G.P is given by

$\frac{a}{1-r}$ (a=first term,r=common ratio)

Here a=1,r=x

$S(1-x)=\frac{1}{1-x}$

$S=\frac{1}{(1-x)(1-x)}$

$\Rightarrow S=\frac{1}{(1-x{)}^2}$