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Home » First, compute $ (x + y)^3 $: - DNSFLEX

First, compute $ (x + y)^3 $: - DNSFLEX

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 We are given the recursive sequence defined by \( b_1 = 1 \) and \( b_{n+1} = f(b_n) = b_n - \frac{b_n^5}{5} \). We aim to find all positive integers \( n \) such that \( b_n < 0.1 \).

📌 Step 1: Analyze the function \( f(u) \)

📌 Note that for \( u > 0 \),

📌 f(u) = u - \frac{u^5}{5} < u,

📌 so the sequence \( \{b_n\} \) is strictly decreasing as long as \( b_n > 0 \).

📌 Step 2: Compute initial terms numerically

📌 We compute successive values:

📌 \( b_2 = 1 - \frac{1^5}{5} = 1 - 0.2 = 0.8 \)

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