Question

If sum of the series $\sum _{n=0}^{\infty }{r}^n=S$ for $\left|r\right|<1$, then sum of the series $\sum _{n=0}^{\infty }{r}^{2n}$ is

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

Answer 1

The correct option is D $\frac{S^2}{2S-1}$

Given, $\sum _{n=0}^{\infty }{r}^n=S$
$\Rightarrow 1+r+r^2+...+\infty =S$
$\Rightarrow \frac{1}{1-r}=S\Rightarrow r=\frac{S-1}{S}$
Now, $1+r^2+r^4+...+\infty =\frac{1}{1-r^2}$
$=\frac{1}{1-{\left(\frac{S-1}{S}\right)}^2}=\frac{S^2}{S^2-{(S-1)}^2}$
$=\frac{S^2}{S^2-(S^2+1-2S)}=\frac{S^2}{2S-1}$