If sum of the series $\sum_{n = 0}^{\infty}r^{n} = S$ for $|r|

Question

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

Dr. Abubacker · Accepted Answer

(d)
Since, $1 + r + r^{2} + ...\infty = S$
$	herefore\ \frac{1}{1 - r} = S\ \  \Rightarrow \ r = \frac{S - 1}{S}$ …(i)
Now, $1 + r^{2} + r^{4} + ...\infty = \frac{1}{1 - r^{2}} = \frac{1}{1 - \left( \frac{S - 1}{S} ight)^{2}}$ [from Eq. (i)]
$= \frac{S^{2}}{S^{2} - (S - 1)^{2}}$
$= \frac{S^{2}}{(2S - 1)}$

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

Explanation

Reviewed by Dr. Abubacker

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