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neck and shoulder stretches - Feedz API

📅 May 18, 2026 👤 scraface

May 18, 2026

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📌 Question: Define $ L(u) = u - \frac{u^3}{3} $ for every real number $ u $. If $ n $ is a positive integer, define $ a_n $ by

📌 a_n = \sum_{k=1}^{n} L\left(\frac{k}{n}\right)

📌 Find $ \lim_{n \to \infty} a_n $.

📌 L(u) = u - \frac{u^3}{3}, \quad a_n = \sum_{k=1}^{n} L\left(\frac{k}{n}\right)

📌 \lim_{n \to \infty} a_n = \lim_{n \to \infty} \sum_{k=1}^{n} \left( \frac{k}{n} - \frac{1}{3} \left( \frac{k}{n} \right)^3 \right)

📌 This is a Riemann sum. Split the sum:

📌 a_n = \sum_{k=1}^{n} \frac{k}{n} - \frac{1}{3} \sum_{k=1}^{n} \left( \frac{k}{n} \right)^3

📌 = \frac{1}{n} \sum_{k=1}^{n} k - \frac{1}{3n^3} \sum_{k=1}^{n} k^3

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