A problem of infinite series

Algebra Level 3

132+1+142+2+152+3++1(n+2)2+n+=? \large \dfrac{1}{3^2+1}+\dfrac{1}{4^2+2}+\dfrac{1}{5^2+3}+ \cdots + \dfrac1{(n+2)^2 + n} + \cdots = \, ?

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