# Fourier Series Help

## Homework Statement

Hello,
Check each function to see whether it is piecewise smooth. If it is, state the value to which its Fourier series converges at each point x in the given interval and the end points

(a.) f(x)=|x|+x, -1<x<1
(it would be very helpful to see if i did this right, as the professor I have does not do examples and that is how I learn how to approach and solve problems)

## Homework Equations

If f is piecewise smooth and is periodic with a period of 2a, then at each point x in the corresponding Fourier series to f converges and its sum is:

Fourier series= 0.5(f(x+)+f(x-)), where f(x+) is the limit from the right, and f(x-) is the limit from the left.

Criterion for piecewise smooth on interval a<x<b:
1) f is piecewise continuous (it is bounded and is continuous, except possibly for a finite number of jumps and removable discontinuities)
2)f'(x) exists except possibly at a finite number of points
3) f'(x) is piecewise continuous

## The Attempt at a Solution

After sketching the function, it is continuous on the interval, f'(x) exists and it has a finite number of discontinuities (so it is piecewise continuous) Therefore, f(x) is piecewise smooth

f(x)= 2x, 0<x<1
0, -1<x<0 (spilt it up)

f(x)=.5(f(x+)+f(x-))=2x/2=x

endpoints: at x=-1 .5(f(-1+)+f(-1-))=-2/2=-1

at x=1, .5(f(1+)+f(1-))=2/2=1

Is this right? I think i went wrong somewhere. And do i have to actually find the Fourier series (which I know how to do, just thought it was not needed/was not even specified)?

## The Attempt at a Solution

Last edited:

ehild
Homework Helper
Your work looks correct, and no need to determine the Fourier series.

ehild

Thank you. I do have other problems but I am going to start fiending in the math learning center for this class. Professor expects us to do problems while showing no examples, and Im not the only one who has this problem in the class.

Once again thank you !!

ehild
Homework Helper