Question

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

$\frac{S^2}{2S-1}$

Explanation for the correct option:

Step1: Calculate the common ratio for the given series.

$\begin{array}{c}S={\sum }_{n=0}^8{r}^n\\ =1+r+r^2+...r^8\\ =\frac{1}{1-r}\\ \Rightarrow r=\frac{S-1}{S}\end{array}$

Step2: Substitute the value from the second series for the value from the first series.

$\begin{array}{c}S={\sum }_{n=0}^8{r}^{2n}\\ =1+r^2+r^4+...r^8\\ =\frac{1}{1-r^2}\\ =\frac{1}{\left(1-\frac{{\left(S-1\right)}^2}{S^2}\right)}\\ =\frac{S^2}{2S-1}\end{array}$

Hence option(4) is the answer