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Home » Yellow + Black: The Secret Neighborhood that’ll Make You Stop Screaming! - Dachbleche24

Yellow + Black: The Secret Neighborhood that’ll Make You Stop Screaming! - Dachbleche24

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Question: Define $ f(u) = u - \frac{u^4}{4} $. If $ n $ is a positive integer, define $ b_n $ by the recurrence

📌 b_1 = 1, \quad b_{n+1} = f(b_n).

📌 Determine the limit $ \lim_{n \to \infty} b_n $.

📌 Solution: We are given the recurrence

📌 b_1 = 1, \quad b_{n+1} = f(b_n), \quad \text{where } f(u) = u - \frac{u^4}{4}.

📌 We aim to find $ \lim_{n \to \infty} b_n $.

📌 First, observe that $ f(u) < u $ for all $ u > 0 $, since $ \frac{u^4}{4} > 0 $. Thus, the sequence $ b_n $ is strictly decreasing and bounded below by 0. Therefore, it converges to some limit $ L \geq 0 $.

📌 Taking limits on both sides of the recurrence:

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