# Homework Help: First nonzero terms of Fourier sine series

1. Dec 17, 2012

### frenchkiki

1. The problem statement, all variables and given/known data

Consider the function φ(x) ≡ x on (0, l). Find the sum of the first three (nonzero) terms of its Fourier sine series.

Reference: Strauss PDE exercise 5.1.3

2. Relevant equations

3. The attempt at a solution

I have found the coefficient without difficulty, it is A$_{n}$= -2l/n$\pi$ cos (n$\pi$) + 2/n$^{2}$$\pi$$^{2}$ sin (n$\pi$).

When I plug in n=1,2,3; I obtain 2l/$\pi$, -2l/$\pi$ and 2l/3$\pi$, respectively.

However the answer is given by: 2l/$\pi$ sin ($\pi$x/l) - 2l/$\pi$ sin (2$\pi$x/l) + 2l/3$\pi$ sin (3$\pi$x/l)

What I found comes from A$_{n}$ for n=1,2,3 but I do not understand which those sine terms come from.

2. Dec 17, 2012

### Dick

So far you've calculated the coefficients of the sine series. Even saying 'coefficients' means they should multiply something. The sum of the sine series is a function of x. The thing the coefficients multiply is exactly those sine functions your are integrating against. Look at equation 6 here http://mathworld.wolfram.com/FourierSeries.html

3. Dec 17, 2012

### frenchkiki

So that would be sin (n$\pi$x/l) for each corresponding n? Do I need multiply the coefficients by sine because I am asked for a sine series?

4. Dec 17, 2012

### Dick

Yes, it's asking you for the sine series. Not just the coefficients.

5. Dec 17, 2012

### frenchkiki

Alright, so if I get this right, to find the first 3 nonzero terms, I plug in my results for n=1,2,3 in equation 6 and disregard the A$_{n}$ cos (nx) terms since I am asked for the sine series, and A$_{0}$ vanishes so I'm left with B$_{n}$ sin (nx) (A$_{n}$ in my answer). Thanks again