Flashcards
Series
Cards in this deck (16)
- 1What is a series?Sum of terms in a sequence (e.g. 1 + 3 + 5 + ...)
- 2Tests of convergenceRatio test, integral test, comparison test → else, seek further help
- 3Ratio test
- 4Integral test
- 5Comparison test
- 6What is a power series?Where aₙ is a sequence of real numbers, a power series is a function defined as an infinite series of form:
f(x) = a₀ + a₁x + a₂x² + ... + aₙxⁿ - 7Radius of convergence of a power seriesR⁻¹ = limₙ→∞ |aₙ₊₁ / aₙ| provided that this limit exists
If R = ∞, series converges all the time. - 8Taylor polynomial of degree n around af(x) = f(a) + f '(a)(x-a) + f ''(a)(x-a)²/ 2! + ... + f⁽ⁿ⁾(a) (x-a)ⁿ/n! + error term/ remainder
- 9Maclaurin seriesA special case of the Taylor series, where a = 0
- 10Fourier series
- 11Fourier coefficients for even function
- 12Fourier coefficients for odd functionSince a₀ = 0 and aₙ = 0
- 13Gibbs phenomenonAs more terms are added to Fourier series sum, approximation gets closer to original function, but there's always an overshoot at point of discontinuity. This overshoot doesn't fade out as no. of terms increases but rather approaches a finite value.
- 14Dirchlet conditions1) f(c) is defined except possibly at a finite number of points in (-L, L), such as discontinuities
2) f(x) is periodic with period 2L
3) f(x) and f'(x) are piecewise continuous in (-L, L) - 15Fourier convergence theorem1) if f is continuous at a point then F(x) = f(x)
2) if f is discontinuous at a point, then Fourier series F(x) converges to middle of the gap:
F(x) = 1/2 (f(G+) + f(G-)) - 16Parseval's theorem
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- Mathematics Flashcards on Series, created by Paris Hart on 19/01/2021.
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- Paris Hart
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