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Home » Since $ t $ must be a whole number, round up to the next month: - Esdistancia

Since $ t $ must be a whole number, round up to the next month: - Esdistancia

📅 March 17, 2026 👤 scraface

Mar 17, 2026

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

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

📌 \frac{1}{2} \left[ \left( \frac{1}{1} - \frac{1}{3} \right) + \left( \frac{1}{2} - \frac{1}{4} \right) + \left( \frac{1}{3} - \frac{1}{5} \right) + \cdots + \left( \frac{1}{50} - \frac{1}{52} \right) \right]

📌 Most terms cancel. The surviving positive terms are \(\frac{1}{1}, \frac{1}{2}\), and the surviving negative terms are \(-\frac{1}{51}, -\frac{1}{52}\) (since \(k+2 = 51, 52\) have no positive counterparts to cancel them).

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