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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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Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. Titles should be informative. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. If this is homework, please add the [homework] tag; people will still help, so don't worry. Also, many find the use of imperative ( "Use") to be rude when asking for help; please consider rewriting your post. –  Arturo Magidin May 4 '12 at 22:01
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If I were to guess, I'd say the homework tag is missing... –  Mariano Suárez-Alvarez May 4 '12 at 22:03
    
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. –  Arturo Magidin May 4 '12 at 22:05
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What's your math question? be specific. –  Argon May 4 '12 at 22:10
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In the last integral I believe there should be $f(x)^2$ instead of $f(x^2)$... –  AD. May 4 '12 at 22:10

1 Answer 1

Hints:

  1. What does even or odd mean? Can you simplify $f(-x)$? Try! (Edit: The function $f$ of the post has changed and the answer too - now look at $f(-x)$ for $x$ close to 0).

  2. How do you calculate Fourier coefficients? These are all Fourier coefficients (you might wish to use the Weierstrass M-test in order to justify interchange summation and integration).

  3. What does Plancherel's (or rather Parseval's actually) formula say for this $f$? (You will end up summing a geometric series.)


I hope you manage to walk through the problems now...

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