Question

The sum of the series $1-3x+5x^2-7x^3+\dots \infty$, when $|x|<1$ is

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

Answer 1

The correct option is C $\frac{1-x}{(1+x{)}^2}$

Let $S_n$ denote the sum of the given infinite series. Now,
$S_{\infty }=1-3x+5x^2-7x^3+\dots \infty$
Here, $a=1,r=-x$ and $d=2$
$\therefore S_{\infty }=\frac{1}{1-(-x)}+\frac{2(-x)}{[1-(-x){]}^2}$
$=\frac{1}{1+x}-\frac{2x}{(1+x{)}^2}$
$=\frac{1-x}{(1+x{)}^2}$