I am not sure how to google the answer for this question. Anyway, in trying to compute the velocity of a charged particle in an electromagnetic field, I came across these two infinite series (Maclaurin series, generating functions):
$$F_1(x) = \displaystyle\sum_{k=0}^{\infty} \displaystyle\frac{(-1)^kx^k}{(2k+1)!}$$
$$F_2(x) = \displaystyle\sum_{k=0}^{\infty} \displaystyle\frac{(-1)^kx^k}{(2k+2)!}$$
Obviously these are similar to sine and cosine, which is what I thought they were at first until I noticed that the exponents were wrong.
What I Have Tried: I think we have that $$xF_1(x^2)=\sin(x)$$
Does this imply that $$F_1(x)=\frac{\sin(\sqrt{x})}{\sqrt{x}}?$$
I think so, but I feel like this "proof" is too hand-wavey. For $F_2(x)$, I suspect there is a similar relationship to cosine, but I'm not sure about how to get the factorials to behave.
EDIT: I think for $F_2(x)$ we have $$x^2F_2(x^2)=1-\cos(x)$$ $$\implies F_2(x) = \frac{1-\cos(\sqrt{x})}{x}$$