The question:
Evaluate the series $$ 1 + \frac 19 + \frac 1{25} + \frac 1{49} + \cdots $$
All the information given is that it is in relation to a Fourier series and that the series is to be evaluated.
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Ben Grossmann
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This answer computes a similar sum using Fourier series, perhaps you can get an idea from there – Ben Grossmann Jul 03 '18 at 14:51
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Where would I begin? – Bjorn MacDonnacha Jul 03 '18 at 14:55
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I don't really know how you should begin. Another idea I have is that the instructor might have a trick in mind where you plug in to some Fourier series formula that you have. Do you have some table of Fourier series? Did you compute the Fourier series of a triangle wave in an earlier problem, perhaps? – Ben Grossmann Jul 03 '18 at 15:00
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Yes, one of my previous asked questions on this site – Bjorn MacDonnacha Jul 03 '18 at 15:49
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Perfect. Once you have the answer to the question you're referring to, it will be sufficient to plug in $x = 0$. – Ben Grossmann Jul 03 '18 at 16:29
2 Answers
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One approach is as follows:
Let $f(t)$ denote a triangle wave with the precise form described here (we'll take $L = 1$). As you can see in the link, $f$ can be expanded into its Fourier series as $$ f(t) = \frac{8}{\pi^2} \sum_{n=1}^\infty \frac{(-1)^{n-1}}{(2n-1)^2} \sin ((2n - 1) \pi t) $$ Now, just plug $t = \frac 12$ into both sides of the equation above.
Ben Grossmann
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Hint: On way to evaluate your series is using Parseval theorem. At first you need to find the fourier series of a function like $f(x)=x$ in $[-\pi,\pi]$ and then applying Parseval theorem.
In our example, the function is odd and you should find the series of it.
Nosrati
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Also here is a ref: https://math.stackexchange.com/questions/1437102/how-to-find-11-221-32-using-fourier-series?rq=1 – Nosrati Jul 03 '18 at 15:30
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