Exploring the Kinetic Energy Formula: Theorem and Definitions

[H.B.]

This theorem (if it is proven) comes also from a formula of kinetic energy.
Has anyone some suggestions about this one.

Definition of f(n):

$$\ f(n)=(2^{n-1}a)Mod(a+b)$$

Definition of g(n):

$$\ \ g(n)= 4f(n)-2(a+b)+1$$

Definition of h(n):

$$\ h(n)= Sign(f(n))(Sign(g(n))+1)$$

Theorem:

$$\lim_{n\rightarrow\infty}\sum_{k=1}^{n}{h(k)\left(\frac{1}{2}\right)^{k+1}} = \frac{a}{(a+b)}$$

Thank you for trying.

 

[H.B.]

n, a and b are integers > 0