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Home » Error rate halves weekly: geometric sequence with ratio 0.5 - Deep Underground Poetry

Error rate halves weekly: geometric sequence with ratio 0.5 - Deep Underground Poetry

📅 May 4, 2026 👤 scraface

May 04, 2026

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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...
\lim_{n \to \infty} a_n = \lim_{n \to \infty} \sum_{k=1}^{n} \left( \f...
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{1}{n} \sum_{k=1}^{n} k - \frac{1}{3n^3} \sum_{k=1}^{n} k^3
Use formulas:
$ \sum_{k=1}^{n} k = \frac{n(n+1)}{2} $
$ \sum_{k=1}^{n} k^3 = \left( \frac{n(n+1)}{2} \right)^2 $
a_n = \frac{1}{n} \cdot \frac{n(n+1)}{2} - \frac{1}{3n^3} \cdot \left(...

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