I wish to find the function that generates the following power series, I wonder if there exists one:
$1-\dfrac{x^2}{2}+\dfrac{x^4}{4}-\dfrac{x^6}{6}+\dfrac{x^8}{8}...$
I know the series expansion for $\cos(x)$ is:
$1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}-\dfrac{x^6}{6!}+\dfrac{x^8}{8!}...$
But I don't know to transform this to obtain the first series.
Is there a way to find the generating function of a given power series? My guess is that not all power series have closed form to which they converge.
If $f(x)=1-\frac{x^2}2+\frac{x^4}4-\frac{x^6}6+\cdots$, then $f'(x)=-x+x^3-x^5+\cdots=-\frac x{1+x^2}$. So, $f(x)=1-\log\left(\sqrt{1+x^2}\right)$.