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ChallengeSuperchallengeAdvanced Number Theory

Advanced Number Theory · Superchallenge problem

Compute ∑k=192k+1k2(k+1)2\displaystyle\sum_{k=1}^{9} \frac{2k+1}{k^2(k+1)^2}. The sum is mn\tfrac{m}{n} in lowest terms; find m+nm + n.