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📰 oxed{2x^2 + x + 1} 📰 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 $. 📰 Rob Lowe Age 56245 📰 Khal Drog0 Exposed The Mysterious Legacy Behind The Legend Shocking Facts Inside 9591089 📰 This Rediscovered Rockport Secret Will Change Everything You Think You Know 5000415 📰 Shocked Banana Plants Multiply Faster Than You Thoughtheres How 4856056 📰 Tanker Truck 5810500 📰 3 Onlyfans Founder Exposed The Hidden Strategy Behind His Content Empire 9232934 📰 Spicy Chat Ai Revealed The Extreme Conversations That Will Blow Your Mind 8912565 📰 Your Stay Just Got A Dramatic Upgrade With Hot Tubs In Every Suiteslip Into Luxury Without Stepping Outside 4097641 📰 The Santa Clause Series 4296218 📰 Koren Grieveson Reveals A Secret No One Expected 4035475 📰 Abby Dalton 4744571 📰 White Corset Hacks Step Into Glam Like Never Beforea Must Try Trend 373204 📰 Jane Of Twilight The Hidden Secrets That Will Blow Your Mind 8000727 📰 Boris And Natasha The Duo That Redefined Love Laughter And Drama 2509636 📰 Funny Emojis 5244160