The fourier coefficients of a 2pi-periodic function are as

Question

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, a_n = (-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, a_n = 1/1+n² and b_n = 1/n³, for all n ≥ 1. Is the function continuous? Justify your answer.