Past exam of the mathematics course of the University of Cambridge / 2014 / ib / Paper 4 / 17D / v

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- Solution v

Solution

For real nonintegral

α

, the Parseval identity and the coefficient formula give

\frac{α ^{2} s i n ^{2} ( π α )}{π ^{2}} \sum_{n \in Z} \frac{1}{( n ^{2} - α ^{2} ) ^{2}} = \frac{1}{2 π} \int_{- π}^{π} cos^{2} (αx) d x = \frac{1}{2} + \frac{s i n ( 2 π α )}{4 π α} .

Solving for the sum yields

n \in Z \sum \frac{1}{( n ^{2} - α ^{2} ) ^{2}} = \frac{π ^{2}}{2 α ^{2} sin ^{2} ( π α )} + \frac{π cot ( π α )}{2 α ^{3}} .

At

α = 5/2

, the sine square is one and the cotangent is zero, giving $2 π^{2} /25$ .

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