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I had to find fourier series for $f(x) = x$, $-\pi < x < \pi$. I found that the Fourier series for $f$ is

$$\sum_{n=1}^{\infty}(-1)^{n+1}\cdot\frac{2}{n}\cdot\sin(nx).$$

Now I have to sketch the graph on $\left[-3\pi,3\pi\right]$. How do I do this?

MattAllegro
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user2013804
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2 Answers2

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The fourier series is infinite, you can only graph a partial sum of the series for your interval. My guess is that it will look something like this in the first 3 terms:

enter image description here

See how the graph is starting to resemble f(x) = x (for x between -Pi and Pi) by the third sum. And also notice that it is periodic with period 2*Pi

MNKY
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By Fourier series , you are representing a function by an infinte sum of sinusoids. You cannot plot all of them. What you can plot is the amplitude of those sinusoids by taking index n on x-axis and magnitude on y-axis.

Watson
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user146334
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  • Could you explain me how exactly this happens? I am very confused. – user2013804 Apr 28 '14 at 12:00
  • Are you asking me how can we represent a function by an infinite sum of sinusoids ? If that is your question i will give you a simple analogy . Consider a vector . A vector can be represented as a linear sum of orthogonal vectors (i,j,k) . The same applies to signals also . But here we extend the orthogonality to infinte number of signals . We define that orthogonal set as a set of functions which obey this property i.e integral (fx.fy) over a ,b is zero for all x,y . This set of sinusoids fall in that category and hence you can break a function into infinite number of sinusoids . – user146334 Apr 28 '14 at 14:53