Engineering Board Exam Reviewer · Fourier Series · medium

The square wave f(x)=-1 on (-π,0) and f(x)=1 on (0,π) has a jump at x=0. To what value does its Fourier series converge at x=0?

1. 1
2. -1
3. The series diverges
4. 0

Answer

D. 0

Solution

At a jump discontinuity the Fourier series converges to the average of the left and right limits: -1+1/2=0.

More Fourier Series questions

1. Find a1 in the Fourier series of f(x)=|x| on (-π,π).
2. Find the constant term a0/2 of the Fourier series of f(x)=x2 on (-π,π).
3. The square wave f(x)=-1 on (-π,0) and f(x)=1 on (0,π) has Fourier coefficients bn. Find b3.
4. Find b3 in the Fourier series of f(x)=x on (-π,π).
5. Which Fourier coefficients vanish for the odd function f(x)=x on (-π,π)?
6. The Fourier series of f(x)=x2 on (-π,π) is π2/3+4 n=1∞(-1)n/n2 nx. Evaluate it at x=π to find n=1∞ 1n2.
7. The half-range sine series of f(x)=x on (0,2) is bn π x/2. Find b2.

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