Question

Sum to $n$ terms the following series:
$1+2x+3x^2+4x^3+...,|x|<1$

Answer 1

Consider the given series.

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

On multiply by $x$ both sides, we get

$Sx=x+2x^2+3x^3+4x^4+.......+n{x}^n$ $...........\left(2\right)$

Subtract equation $\left(2\right)$ from equation $\left(1\right)$, we get

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

$S(1-x)=1+x+x^2+x^3+x^4+......+x^{n-1}+n{x}^n$

So,

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

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

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

Hence, this is the answer.