Find an approximation for Fourier Sine Series

Hello Everyone! Today we are asked to find an an approximation for f(x) that gives a finite sum values for all x where f(x) is defined as $\displaystyle \sum_{n=1}^{\infty}\frac{1}{n}\sin(nx)$. I don't know if I rephrased it right, but the exact problem goes like this:

"*What does the statement* $\displaystyle \sum_{n=1}^{\infty}\frac{1}{n}\sin(nx)$ *mean? Can you write down a ﬁnite sum approximation of f(x)?*"

Since this one resembles a fourier sine series, my first guess is to find the exact function,f(x), such that $\displaystyle \frac{1}{n}=\frac{2}{\pi}\int_{0}^{\pi}f(x)sin(nx) dx$

But from there I'm lost as to how I would find f(x). Anyone got any idea to find f(x) or at least find an approximation of it? Thanks everyone in advance!(Happy)

Re: Find an approximation for Fourier Sine Series

Hey EliteAndoy.

If you were going to get some kind of approximation for f(x), you should not that all the cosine terms are zero which means that f(x) is orthogonal to cos(nx).

What kind of functions are orthogonal to cosines in fourier series integrals/inner products?

Re: Find an approximation for Fourier Sine Series

Just curious..Is this calculus? If so, it's not single-variable calculus is it? What calculus is this? :)

Re: Find an approximation for Fourier Sine Series

Quote:

Originally Posted by

**Paze** Just curious..Is this calculus? If so, it's not single-variable calculus is it? What calculus is this? :)

It's a Fourier series, which is studied in Fourier analysis. This question is in fact single-variable analysis, of which calculus is part of. You can think of analysis and fourier analysis as an extension of calculus for now, though some may argue that Fourier analysis is an application of calculus and other areas of analysis. Perhaps it is a little bit of both.

Re: Find an approximation for Fourier Sine Series

Sorry for late reply. Been working with complex number for the mean time. But I think in an fourier series integral, sine functions are orthogonal to cosine at the interval $\displaystyle |x|\leq \L$ since it would yield to zero as its value. Still I'm not sure as to what specific sine function I should use for f(x). I really have no idea since I've never seen these stuff before as I am only at Calc 2 and its really weird that we got this kind of question for our homework. But yeah, can you give me more hint as to how I should approach this problem? Thanks.(Nod)

Re: Find an approximation for Fourier Sine Series

Think about the integral of sin(ax+b). The integral of this should be -1/a * cos(ax+b). Now relate the a to the variable n.

Re: Find an approximation for Fourier Sine Series

Quote:

Originally Posted by

**EliteAndoy** Hello Everyone! Today we are asked to find an an approximation for f(x) that gives a finite sum values for all x where f(x) is defined as $\displaystyle \sum_{n=1}^{\infty}\frac{1}{n}\sin(nx)$. I don't know if I rephrased it right, but the exact problem goes like this:

"*What does the statement* $\displaystyle \sum_{n=1}^{\infty}\frac{1}{n}\sin(nx)$ *mean? Can you write down a ﬁnite sum approximation of f(x)?*"

Since this one resembles a fourier sine series, my first guess is to find the exact function,f(x), such that $\displaystyle \frac{1}{n}=\frac{2}{\pi}\int_{0}^{\pi}f(x)sin(nx) dx$

But from there I'm lost as to how I would find f(x). Anyone got any idea to find f(x) or at least find an approximation of it? Thanks everyone in advance!(Happy)

Look at a table of Fourier series, you will find something you can use to answer this question in the saw tooth waveform section.

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