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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 $. 📰 Add More Chaos Top 5 Online 3 Player Games You Have To Try Asap 9305702 📰 Bank Of America Warren Nj 8050592 📰 Best Buy Roku Tv 3723291 📰 The Eye Catching Lab German Shepherd Mix Youll Want A Picket Fenced Home In Minutes 4118486 📰 Unlock The Secret Of Ccc One Are You Ready For Life Changing Results 6884547 📰 Show In Spanish 6853960 📰 Download The Best Excel Monthly Calendar Layoutyour Planner Will Thank You 6140686 📰 Why This Simple Trick Is Rewriting The Rules Of Mexican Salads 990987 📰 Wells Fargo Foreign Currency Rates 8527543 📰 Voldemort And 5379744 📰 Student Deal Alert Microsoft Office 50 Offperfect For School Life 717646 📰 Fld Stock Shock Investors Are Losing Millions Over This Hidden Market Flood 8291227 📰 Dr Groot Shampoo Exposed Hidden Power That Transforms Your Grow 2892563 📰 Watch How Rmd Distribution Transforms Supply Chains With Shocking Precision 4680364 📰 Are Ne Patriots Playing Today 8868619