# Fourier - terms of the series as well as f(x vs x for for L...

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MATH 342-010 Differential Equations with Linear Algebra, November 29, 2009 Fourier Series The Fourier series for the function f(x)=(x −L) 2 (x + L) 2 is given by f(x)= 8L 4 /15 48L 4 4 1 𝑛 = 𝑛 =1 n cos(nπx/L) Note that f(x)is smooth in its domain. Above are plotted the function f (thickest line), as well as the Fourier series taking the first one, two, and three terms in the sum

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In increasing order of thickness: Fourier series keeping the first one, two and three
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Unformatted text preview: terms of the series as well as f(x) vs x for for L = 1 Remarks 1. Note that with each increasing term, the series becomes a better approximation. 2. If we take the frst N terms of the series, the absolute value of the (N +1) st term is a rough estimate of the error. Note this term is always smaller than 48L 4 /(N+1) 4 π 4 which decays very rapidly as N gets large....
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