What does Plancherel's (or rather Parseval's actually) formula say for this f?

Question

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?

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.)

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