What is the period of this function?

[pivoxa15]

$$f(x) = \sum_{n=0}^\infty \frac{cos(4^nx)}{3^n}$$

Usually the period T is worked out by doing $$\frac{2\pi}{w}=\frac{T}{n}$$ where w is the angular frequency and T is the period. n is the function number in the series. But it obviously can't be applied here.

 

[0rthodontist]

Look at the frequency of cos(4^n x), and make a guess.

 

[pivoxa15]

I realized that the cosine function with the longest period will dictate the period of the series because the other terms will all have less periods but all have an integer number of periods during the interval of the largest period. In my case, the series will have a period of $$2\pi$$.

 

[pivoxa15]

What kind of function is f(x) what does it represent? It does not seem to be a normal Fourier series because its angular frequency is not an intger multiple of n but a power.

 

[StatusX]

It's a Fourier series. It's just that most of the a_n are zero.

 

[pivoxa15]

Could you predict what f(x) is in terms of a non infinite series?

 

[StatusX]

It doesn't have a closed form. It's similar to a http://en.wikipedia.org/wiki/Weierstrass_function" , so it is continuous but nowhere differentiable.