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Q. The sum $\displaystyle\sum_{x=0}^{\infty} \frac{1}{x^2+5 x+4}$ is equal to

Sequences and Series

Solution:

$\frac{1}{3} \displaystyle\sum_{ x =0}^{\infty}\left(\frac{1}{ x +1}-\frac{1}{ x +4}\right)$
Substituting $0,1,2,3, \ldots \ldots \infty$, we get $=\frac{11}{18}$