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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 $. 📰 Watchmen 2009 Cast 5266729 📰 How Much Liquid Can You Take On A Plane 4387022 📰 Futura Font 609372 📰 Hungry And Tired The Ultimate Snackpass Guide To Snaging Free Snacks All Day Long 8542022 📰 A Car Accelerates Uniformly From Rest To A Speed Of 30 Ms In 10 Seconds What Is The Acceleration Of The Car 4799437 📰 Unlock Massive Storage The Ultimate Guide To Windows 11 Thumb Drives You Cant Ignore 474651 📰 How To Master Keyboarding Without Tears In Just 7 Days 5555145 📰 5Mediawize See Your Pcs Secretsclick Now To Find Out 7139550 📰 Go Stock Now The Secret Hacks Experts Say You Need To Know Today 5189693 📰 How A Belt Buckle Knife Turned Into The Hottest Accessory Trendshocking Style Revealed 9441123 📰 Force America 5158562 📰 Wall Furnishing Ideas 2325059 📰 Drink Bottle Brands 8955420 📰 Discover The Secret Hack To Edit Drop Down Lists Like A Pro In Excel 7981391 📰 Support Fios Tv Sign In Activate 1002373