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📰 Question: Define $ L(u) = u - rac{u^3}{6} $ for all real $ u $. If $ n $ is a positive integer, define $ a_n $ by $ a_1 = rac{\pi}{2} $, $ a_{n+1} = L(a_n) $. Find $ \lim_{n o \infty} a_n $. 📰 Solution: The recurrence $ a_{n+1} = a_n - rac{a_n^3}{6} $ resembles the Taylor series for $ rctan(u) $, where $ rac{d}{du} rctan(u) = rac{1}{1 + u^2} $. However, the recurrence is not exact. Assume the limit $ L $ exists. Then $ L = L - rac{L^3}{6} \Rightarrow rac{L^3}{6} = 0 \Rightarrow L = 0 $. To confirm convergence, note $ a_1 = \pi/2 pprox 1.57 > 1 $, and $ a_{n+1} = a_n(1 - rac{a_n^2}{6}) $. Since $ a_1 < \sqrt{6} $, $ a_n $ is decreasing and bounded below by 0. By monotone convergence, $ a_n o 0 $. 📰 Question: Find the center of the hyperbola $ 4x^2 - 12x - 9y^2 + 18y = 27 $. 📰 Jousinns Secret Thatll Make You Blow Your Mindcan You Handle It 7811387 📰 Bank America Rewards 2168829 📰 You Wont Believe How Crazy The Craze Game Took Over The Worldplay Now 2803381 📰 From Chaos To Clarity 5 Proven Project Planning Templates That Work 6439331 📰 Microsoft Dynamics Implementation Partners 9943987 📰 Craving Faster Results Bachilles Is Changing The Game Watch Now 9026021 📰 Superando El Papel Por Qu Las Personas Eligen Lo Tangible Sobre Lo Escrito 9377471 📰 Download These Viral Feliz Dia Quotes To Brighten Your Mood Instantly 2730268 📰 Newtons Units 5979358 📰 The Secret Love Kennedy Never Talked About Was The Moment He Vanished Without Saying Goodbye 7383222 📰 These Wide Toe Box Shoes Are Taking Over The Marketno More Blisters Or Sore Feet 7682749 📰 Deion Sanders Forty Yard Dash 7500653 📰 When Do The Clocks Change In The United States 5900338 📰 Prismatic Evolution Pre Order Alert Systems Going Wild Secure Your Spot Now 8934570 📰 Stacked Jeans The Game Changing Trend Everyones Swooning Over Exclusive Info Inside 4995748