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I'm 18 years old and this is my first time writing a question asking for help. So I hope people out there can really help me by not just giving me tips only. My question is how to solve this fourier series ?

Given that

$$f(t) = \begin{cases}50, &\text{for }t\in(0,2) ;\\ 100 , &\text{for }t=2.\end{cases}$$

Compute the fourier series.

I like to solve Maths question but I end up I can't resolve it because I don't have proper reference book on this topic and my teacher doesn't teach this topic in detail. So guys at least give some work done so that I able to understand clearly.

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    The definition of $f(t)$ is not clear to me, but if $f(t)$ is constant in its domain of definition except for the value at one single point, then the Fourier series has only one nonzero term, which is that constant.2012-09-04
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    Sir could you please show me the wok done on how to solve it.I'm beginner in math and i told earlier that I only have few basics on fourier series. This is the link for the question https://docs.google.com/file/d/0B9YdEFvHB3a9U3FSX2UzWGE5WXM/edit2012-09-04

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The Fourier series of a function $f(t)$ defined on the interval $0 is an expression $${1\over 2}a_0+\sum_{n=1}^\infty \bigl(a_n\cos(n\pi t)+b_n\sin(n\pi t)\bigr)$$ where the coefficients $a_n$ and $b_n$ are given by $$a_n=\int_0^2f(t)\cos(n\pi t)\,{\rm d}t,\qquad n=0,1,2,\dots$$ $$b_n=\int_0^2f(t)\sin(n\pi t)\,{\rm d}t,\qquad n=1,2,3,\dots$$ When you substitute your function $f(t)$ into the expressions above and compute the coefficients, you will find that the value $f(2)=100$ does not matter, since changing a function at only one point will not change the value of the integral. ("The area under a point is zero".) The integrals are therefore $$a_0 =\int_0^2 50\,{\rm d}t=100$$ $$a_n=\int_0^2 50\cos(n\pi t)\,{\rm d}t={1\over 50n\pi}\sin(n\pi t)\Bigr|_0^2 =0, \qquad n=1,2,3,\dots$$ $$b_n=\int_0^2 50\sin(n\pi t)\,{\rm d}t=-{1\over 50n\pi}\cos(n\pi t)\Bigr|_0^2 =0, \qquad n=1,2,3,\dots$$

This means you have a rather "boring" Fourier series, since all terms vanish with the exception of one term, and that's why I wondered if the function $f(t)$ was given correctly.

You then have the Fourier series $${1\over 2}a_0=50$$ which will "converge" to $50$ for all $t$. In particular, it will not converge to $f(t)$ when $t=2$.

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    Thank you very much Per Manne for your great explanation and work done on it. Nw you see, I able to do fourier series without any problem using your answer as guide. Now I know how to compute the fourier series for " |cosx| for all x ". But the question asking again " which values of x, does the series fail to converge to ? To what values does it converge at those points? ". So what is the answer ??2012-09-05
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    This question I get it from this site posted by garett in think. This is the link http://math.stackexchange.com/questions/190535/how-to-solve-fourier-series-for-lvert-cos-x-rvert?rq=12012-09-05