# Question on fourier series

• Dec 8th 2011, 05:38 AM
NFS1
Question on fourier series
Hi,

Im trying to attempt a few questions for an exam in a couple of weeks, but am finding it hard, so was wondering if anyone on here is able to help. Below is the question:

1. The function f(x) is periodic with period and is defined by

f(x) = {-cos x when -pi <x<0
{cos x when 0<x<pi

a) Sketch f from x=-3pi to x=3pi .

b) Decide if f is an even, odd function or neither.
c) Find the Fourier series of f.
d) By choosing a suitable value of x sum the series

1/1.3 - 3/5.7 + 5/9.11 - 7/13.15 + .........
• Dec 8th 2011, 07:39 AM
TheEmptySet
Re: Help needed to answer question on fourier series
Quote:

Originally Posted by NFS1
Hi,

Im trying to attempt a few questions for an exam in a couple of weeks, but am finding it hard, so was wondering if anyone on here is able to help. Below is the question:

1. The function f(x) is periodic with period and is defined by

f(x) = {-cos x when -pi <x<0
{cos x when 0<x<pi

a) Sketch f from x=-3pi to x=3pi .

b) Decide if f is an even, odd function or neither.
c) Find the Fourier series of f.
d) By choosing a suitable value of x sum the series

1/1.3 - 3/5.7 + 5/9.11 - 7/13.15 + .........

for part a) just graph the piecewise function and shift it right and left using the fact that it is periodic.

For b)

look at two different cases

Case 1 $\displaystyle x > 0$ then

$\displaystyle f(-x)=-\cos(-x)=-\cos(x)=-f(x)$ so it is odd so far....

Now you try case 2

$\displaystyle x < 0$

From part b) you will find that the function is odd!

So you only need to compute half of the fourier coefficents. Which ones? Use this to get started and post back with some work if you get stuck.