How can I visualise $f(k,x)=\sum\limits_{n=0}^{\infty}\frac{x^{kn+1}}{(kn+1)!}(-1)^n$?

Question

We can be sure, that $$\sin(x)=\sum\limits_{n=0}^{\infty}\frac{x^{2n+1}}{(2n+1)!}(-1)^n$$ And, of course, $\sin(x)$ relates with circle. Which figure or infinite curve generates $$f(k,x)=\sum\limits_{n=0}^{\infty}\frac{x^{kn+1}}{(kn+1)!}(-1)^n$$ for example, when $k=3$? How can I visualise it?