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Q.
The sum to infinite terms of the series $\frac{1}{1-\frac{1}{4}}+\frac{1}{(1+3)-\frac{1}{4}}+\frac{1}{(1+3+5)-\frac{1}{4}}+\ldots \ldots . . \infty$ is
Sequences and Series
Solution:
$ t _{ n }=\frac{1}{1+3+5+7+\ldots .+(2 n -1)-\frac{1}{4}} $
$=\frac{1}{ n ^2-\frac{1}{4}}=\frac{4}{4 n ^2-1}=\frac{4}{(2 n -1)(2 n +1)}=2\left\{\frac{1}{2 n -1}-\frac{1}{2 n +1}\right\} $
$\therefore S _{ n }=\Sigma t _{ n }=2\left\{\frac{1}{1}-\frac{1}{2 n +1}\right\}=\frac{4 n }{2 n +1} .$