Taylor Series are Limits of Legendre Expansions - Gvsu - Grand ...

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The Proof • By Taylor’s Theorem, f(th) = f(0)+f ′ (0)th+...+ f(n) (0) n! (th) n +O ( h n+1) as h → 0. • For each 0 ≤ m ≤ n − 1, t m is orthogonal to P n . • 2n+1 2 ∫ t=1 t=−1 t n P n (t)dt = 2n (n!) 2 (2n)! • P n (z/h) = 1 ( ) (2n)! h n 2 n (n!) 2 · zn + O(h) as h → 0. Taylor Series areLimits of Legendre Expansions – p.10/15
The Proof Combining all these ideas in just the right way yields ( ∫ ) 2n + 1 t=h lim f(t)P n (t/h)dt P n (z/h) = f(n) (0) h→0 + 2h n! for each n ≥ 0. t=−h Taylor Series areLimits of Legendre Expansions – p.11/15
Page 1 and 2: Taylor Series are Limits of Legendr
Page 3 and 4: A Common Question in Calculus Can w
Page 5 and 6: Taylor’s Theorem If f is analytic
Page 7 and 8: Neumann’s Theorem (1862) If f is
Page 9 and 10: Legendre Expansions on [−h, h] If
Page 11 and 12: The Limit of the Legendre Expansion
Page 13 and 14: The Limit of the Legendre Expansion
Page 15 and 16: The Proof Assume first that we may
Page 17 and 18: The Proof • By Taylor’s Theorem
Page 19: The Proof • By Taylor’s Theorem
Page 23 and 24: Bounding the Series Terms The key i
Page 25 and 26: Bounding P n (z/h) To estimate |P n
Page 27 and 28: Bounding the Series Terms Applying
Page 29: Acknowledgements • Nathanial Burc

Taylor Series are Limits of Legendre Expansions - Gvsu - Grand ...

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