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📅 March 1, 2026 👤 scraface

Mar 01, 2026

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We are to compute:
\sum_{k=1}^{50} \frac{1}{k(k+2)}
We use partial fraction decomposition:
\frac{1}{k(k+2)} = \frac{A}{k} + \frac{B}{k+2}
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( \...

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