Question

The sum of the infinite series $1^2+2^{2}x+3^2{x}^2+$........ is.

• A. $\frac{1+x}{(1-x{)}^3}$
• B. $\frac{1+x}{1-x}$
• C. $\frac{x}{(1-x{)}^3}$
• D. $\frac{1}{(1-x{)}^3}$

Answer 1

The correct option is A $\frac{1+x}{(1-x{)}^3}$

Let $S_{\infty }=1+2^{2}x+3^2{x}^2+4^2{x}^3+....\infty$
$S_{\infty }=1+4x+9x^2+16x^3+....\infty$ .... (1)
$\therefore x{S}_{\infty }=0+x+4x^2+9x^3+....\infty$ .... (2)
Subtracting (2) from (1), we get
$(1-x)S_{\infty }=1+3x+5x^2+7x^3+....\infty$ .... (3)
Again $x(1-x)S_{\infty }=0+x+3x^2+5x^3+....\infty$ .... (4)
Subtracting (4) from (3), we get
$(1-x)(1-x)S_{\infty }=1+2x+2x^2+2x^3+....\infty$
$(1-x{)}^2{S}_{\infty }=1+\frac{2x}{1-x}$ $=\frac{1+x}{1-x}$
$\Rightarrow S_{\infty }=\frac{(1+x)}{(1-x{)}^3}$
Hence, option 'A' is correct.