Generating function of $1 + x^k + x^{2k} +\cdots$

Question

I know that the generating function of $1 + x + x^2 + \cdots$ is $\frac{1}{1 - x}$. But what is the generating function of $1 + x^k + x^{2k} + x^{3k} + \cdots$ ?

Answer 1

If $$\frac{1}{1-x} =1+x+x^2+x^3+x^4+\ldots, \tag{$|x|<1$}$$ then $$\frac{1}{1-x^k} =1+(x^k)+(x^k)^2+(x^k)^3+(x^k)^4+\ldots \tag{$|x^k|<1$}$$ Note that $|x|<1$ and $k\in \mathbb{N}$ implies $|x^k|<1$.