I need to find the sum of the Series

Evaluate ` sum_(n=1)^(infty)(2n+1)/(n^2(n+1)^2) `

Consider the sequence of partial sums:

n=1 3/4

n=2 3/4+5/36=8/9

n=3 3/4+5/36+7/144=15/16

n=4 3/4+5/36+7/144+9/400=24/25

The mth partial sum is given by ` 1-1/(m+1)^2 `

Since the limit of this expression as m grows without bound is 1, this will be the sum.

Thus the sum is 1.

Comments

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