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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 $. 📰 Home Security Systems Pricing 📰 Os X Soundflower 📰 Qvc Com Recently On Air Today 3773442 📰 Why Companies Are Turning To Fidelity Cds These Interest Rates Are Too Good To Ignore 1749030 📰 Nio Option Chain 📰 You Wont Believe How Fast These Oracle Lms Scripts Cut Course Setup Time By 90 4871510 📰 Servbank Review The Hidden Bank Thats Revolutionizing Personal Finances 2133331 📰 3 The Hidden Timeline When Can You Actually Get Medicaid 2015308 📰 Tivoli Lodge Vail 4627889 📰 You Wont Believe The Hidden Gems In Pokemon 3Ds Games Revealed 8013616 📰 Why Aap Stock Suddenly Backsteep Cargo 1416431 📰 Co Ed Killer 7699 📰 5 G Home Internet 📰 Binary To Hex 3895660 📰 Fortnite Server Errors