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Q.
If the sum of the series $\displaystyle\sum_{n=0}^{\infty} r^{n},|r|<1,$ is $s,$ then sum of the series $ \displaystyle\sum_{n=0}^{\infty} r^{2 n}$ is
Sequences and Series
Solution:
$ s = \displaystyle\sum_{n=0}^{\infty} r^{n}=1+r+r^{2}+r^{3}+\cdots \text { to } \infty $
$=\frac{1}{1-r} $
$\Rightarrow r=1-\frac{1}{s}=\frac{s-1}{s}$
$ \displaystyle\sum_{n=0}^{\infty} r^{2 n}=\frac{1}{1-r^{2}}=\frac{1}{1-\frac{(s-1)^{2}}{s^{2}}}=\frac{s^{2}}{2 s-1}$