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why is my ear ringing all of a sudden - inBeat

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 Define \( f(u) = u - \frac{u^5}{5} \) for every real number \( u \). If \( n \) is a positive integer, define \( b_n \) by \( b_1 = 1 \) and \( b_{n+1} = f(b_n) \). Find all values of \( n \) for which \( b_n < 0.1 \).

📌 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:

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