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  4. The sum of the series (1/3)+(1/15)+(1/35)+(1/63)+ ldots ldots ldots terms, is

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Q. The sum of the series $\frac{1}{3}+\frac{1}{15}+\frac{1}{35}+\frac{1}{63}+\ldots \ldots \ldots$ terms, is

Sequences and Series

Solution:

$S=\frac{1}{1 \cdot 3}+\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}+\frac{1}{7 \cdot 9}+\ldots \ldots \infty$
$=\frac{1}{2}\left[\left(1-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{5}\right)+\left(\frac{1}{5}-\frac{1}{7}\right)+\left(\frac{1}{7}-\frac{1}{9}\right)+\ldots \ldots \infty=\frac{1}{2}\right.$.