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📅 May 3, 2026 👤 scraface

May 03, 2026

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Multiplying both sides by \(k(k+2)\):
Set \(k = 0\): \(1 = A(2) \Rightarrow A = \frac{1}{2}\)
Set \(k = -2\): \(1 = B(-2) \Rightarrow B = -\frac{1}{2}\)
\frac{1}{k(k+2)} = \frac{1/2}{k} - \frac{1/2}{k+2} = \frac{1}{2} \left...
\sum_{k=1}^{50} \frac{1}{k(k+2)} = \frac{1}{2} \sum_{k=1}^{50} \left(...
\frac{1}{2} \left[ \left( \frac{1}{1} - \frac{1}{3} \right) + \left( \...
Most terms cancel. The surviving positive terms are \(\frac{1}{1}, \fr...
\frac{1}{2} \left( 1 + \frac{1}{2} - \frac{1}{51} - \frac{1}{52} \righ...
Compute \(\frac{1}{51} + \frac{1}{52} = \frac{52 + 51}{51 \cdot 52} =...
\frac{1}{2} \left( \frac{3}{2} - \frac{103}{2652} \right) = \frac{1}{2...

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