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  4. The sum of infinite series (1/9)+(1/18)+(1/30)+(1/45)+(1/63) ldots ldots is equal to

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Q. The sum of infinite series $\frac{1}{9}+\frac{1}{18}+\frac{1}{30}+\frac{1}{45}+\frac{1}{63} \ldots \ldots$ is equal to

Sequences and Series

Solution:

$\Theta \quad S =\frac{1}{9}+\frac{1}{18}+\frac{1}{30}+\frac{1}{45}+\frac{1}{63} \ldots .$
$=\frac{1}{3}\left(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+\frac{1}{15}+\frac{1}{21}+\ldots \ldots\right) $
$=\frac{2}{3}\left(\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\frac{1}{4 \cdot 5}+\frac{1}{5 \cdot 6}+\ldots \ldots\right) $
$=\frac{2}{3}\left(\displaystyle\sum_{n=2}^{\infty} \frac{1}{ n ( n +1)}\right) $
$=\frac{2}{3}\left(\displaystyle\sum_{ n =2}^{\infty}\left(\frac{1}{ n }-\frac{1}{ n +1}\right)\right)=\frac{2}{3} \times \frac{1}{2}=\frac{1}{3}$