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Given $$f(x) = 1+\sum_{n=1}^{\infty}\frac{\sin (nx)}{3^n}$$

what is the easy way to find out the following equation's answer is odd or even?

  • $$\begin{align*} &\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\,dx\\ &\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(3x)\,dx\\ &\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sin(5x)\,dx \end{align*}$$

1) =a_0/2=1/2 odd

2) =0 no cosine terms

3) =1/3^5 =1/243 odd

Sum of odd function is odd

How to calculate following f by using Plancherel's Theorem? or Parseval's theorem? $$\frac{1}{\pi}\int_{-\pi}^{\pi}f\bigl(x^2\bigr)\,dx.$$

this is also given with the question as a Hint- (geometric series formula ∑r^n= r/(1-r), if (r|<1.))

To calculate this by plancherel or Parseval's theorem are we going to use the given function?

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    Please think a bit and give an *informative* title, don't just repeat the instructions you are given (namely, to come up with a specific, informative title!). Please don't just copy your assignment here. Please provide context.2012-05-04
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    The tag (differential-equations) might be misleading, I think (Fourier-series) might be better.2012-05-04
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    What's your math question? be specific.2012-05-04
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    In the last integral I believe there should be $f(x)^2$ instead of $f(x^2)$...2012-05-04
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    @Argon A function is given, and some values are to be calculated.2012-05-04
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    @AD.: The original read: `(1/π) ∫_(-π)^π f((x)^2)dx `; so the OP certainly wrote $f(x^2)$; whether he *meant* it to be $f(x)^2$... well, somehow I suspect we may never know.2012-05-04
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    @ArturoMagidin Thanks, I would guess it was a typo :)2012-05-04
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    @AD. The title used to be "What's your math question? be specific". It was changed.2012-05-04
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    1. First part was just to find the answers are odd or even etc.i think . So when i try to solve the 2) 2nd part of problem . all i see is "use Plancherel's Theorem to calculate (1/π) ∫_(-π)^π f(x)^2dx So i guess we going to use the given function to solve this problem?2012-05-04
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    @andy_Wiz In your comment you say $f(x)^2$ but in the post it is $f(x^2)$ (in which case there is no direct connection to the Plancherel/Parseval thm).2012-05-05

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