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📅 May 18, 2026 👤 scraface

May 18, 2026

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📌 A science policy analyst is evaluating a proposal involving the sum of a series for technological forecasting. Compute \(\sum_{n=1}^{50} \frac{1}{n(n+1)}\).

📌 The series \(\sum_{n=1}^{50} \frac{1}{n(n+1)}\) is a telescoping series. We simplify each term using partial fractions:

📌 Partial fraction decomposition:

📌 \frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}

📌 Write the series using partial fractions:

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

📌 Observe the telescoping nature:

📌 The series expands as:

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