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Q. The value of $\displaystyle\sum_{n=1}^{\infty} \frac{3 n^2+3 n+1}{\left(n^2+n\right)^3}$ equals to

Sequences and Series

Solution:

$\displaystyle\sum_{n=1}^{\infty} \frac{3 n^2+3 n+1}{n^3(n+1)^3}=\displaystyle\sum_{n=1}^{\infty}\left(\frac{1}{n^3}-\frac{1}{(n+1)^3}\right)=1$