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?