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

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Bounding the Series Terms Applying Picone’s result, our previous estimate on the Legendre coefficients, and Stirling’s approximation, we arrive at lim n→∞ ( ∣ 2n + 1 2h ∫ t=h t=−h ) f(t)P n (t/h)dt P n (z/h) ∣ 1 n ≤ 2|z| + 2h R + √ R 2 − h 2. Since |z| < R, this latter quantity is strictly less than 1 for all h sufficiently small, and the proof is complete. Taylor Series areLimits of Legendre Expansions – p.14/15
Acknowledgements • Nathanial Burch, GVSU; • Gisella Licari, GVSU; • Mario Martelli, Claremont McKenna College; • Paul Nevai, Ohio State University; • GVSU Research and Development Center. Taylor Series areLimits of Legendre Expansions – p.15/15
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 and 20: The Proof • By Taylor’s Theorem
Page 21 and 22: The Proof Combining all these ideas
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: Bounding the Series Terms Applying

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

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