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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 $. 📰 Flip Phone Nyt 8083143 📰 This Mexican Hot Dog Recipe Turns Every Taco Night Into A Taste Bomb 8912828 📰 From Blueprint To Budget 2 Story House Plans You Cant Afford To Miss 1270080 📰 Games Free Download Pc 5983720 📰 This Mine Game Will Make You Rich Overnightwatch The Action 3341044 📰 Finally Microsoft Edge For Linux The Ultimate Blend Of Speed Security And Simplicity 141700 📰 All Youll Ever Want In Every Bite Of Steamed Dumplingstry Them Tonight 4369642 📰 A Star Is Born 1976 3402077 📰 How The Best Nail Tech Chairs Actually Make Station Setup Then And There 7336848 📰 Unlock Your Cloud Potential Discover Oracle Cloud Integration Services You Cant Ignore 474917 📰 Fun Games On Roblox 8902325 📰 Double Your Roi Step By Step Guide To Advertising On Bing Start Now 6682832 📰 Winter Weather Shelter Building 3578162 📰 No More Soggy Bagel Bitesair Fry Them To Golden Perfection 8074012 📰 The Ultimate Excel Switch Full Hack Transform Spreadsheets In Seconds 6703720