Question

The sum of $1+2x+3x^2+4x^3+......\infty$ is

• A. $\frac{1}{(1-x)}$
  
• B. $\frac{x}{\left[\right(1-x{)}^2]}$
  
• C. $\frac{1}{(1-x)}+\frac{x}{\left[\right(1-x{)}^2]}$
  
• D. $\frac{1}{\left[\right(1-x{)}^2]}$

Answer 1

The correct option is D $\frac{1}{\left[\right(1-x{)}^2]}$

The given series in an arithmetic -geometric servies whose corresponding A.P. and G.P. are 1,2,3,4,.....
and $1,x,x^2,x^3,$..... respectively. The common ratio of the G.P. is x. Let $S_{\infty }$ denote the sum of the given series.
Then, $S_{\infty }=1+2x+3x^2+4x^3+....\infty ......\left(i\right)\Rightarrow x{S}_{\infty }=x+2x^2+3x^3+.....\infty .....\left(ii\right)$
Subtracting (ii) from (i), we get
$S_{\infty }-x{S}_{\infty }=1+[x+x^2+x^3+....\infty ]\Rightarrow S_{\infty }(1-x)=1+\frac{x}{1-x}\Rightarrow S_{\infty }=\frac{1}{1-x}+\frac{x}{(1-x{)}^2}=\frac{1}{(1-x{)}^2}$