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📅 May 17, 2026 👤 scraface

May 17, 2026

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📌 For $ k = -1 $: $ 1 = B(-1)(1) \implies B = -1 $.

📌 For $ k = -2 $: $ 1 = C(-2)(-1) \implies C = \frac{1}{2} $.

📌 \frac{1}{k(k+1)(k+2)} = \frac{1}{2k} - \frac{1}{k+1} + \frac{1}{2(k+2)}.

📌 The sum becomes:

📌 \sum_{k=1}^{50} \left( \frac{1}{2k} - \frac{1}{k+1} + \frac{1}{2(k+2)} \right) = \frac{1}{2} \sum_{k=1}^{50} \frac{1}{k} - \sum_{k=1}^{50} \frac{1}{k+1} + \frac{1}{2} \sum_{k=1}^{50} \frac{1}{k+2}.

📌 Reindex the sums:

📌 = \frac{1}{2} \left( \sum_{k=1}^{50} \frac{1}{k} \right) - \left( \sum_{k=2}^{51} \frac{1}{k} \right) + \frac{1}{2} \left( \sum_{k=3}^{52} \frac{1}{k} \right).

📌 Combine terms:

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