Reference no: EM131103861
Math 054 Partial Differential Equations - HW Assignment 5
1. Use the Weierstrass M-text to establish the uniform convergence of the given series, on the given interval.
(a) k=1∑∞(cos kx/k2 + sin kx/k3); for all x.
(b) k=1∑∞(x/10)k; |x| ≤ 9.
2. The Fourier coefficients of a 2π-periodic function are as follows: a0 = 0, an = (-1)n/n2 and bn = 1/n2, for all n ≥ 1. Is the function continuous? Justify your answer.
3. Verify that the differential equation
y'' + 4y = n=1∑∞((-1)n+1/n2) sin nπt, (t > 0)
has solution
y(t) = n=1∑∞((-1)n+1/n2(4 - n2π2))sin nπt
by substituting back into the differential equation. Justify all term wise differentiations.
4. The Fourier coefficients of a 2π-periodic function are as follows: a0 = 1, an = 1/1+n2 and bn = 1/n3, for all n ≥ 1. Is the function continuous? Justify your answer.
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