Is there a closed form of $\sum_{n=0}^{\infty} \frac{(-1)^n}{(4n+1)!!}$? This may be an impossible problem.  But I imagine it's worth asking still.
What is the closed form of the sum:
$$\sum_{n=0}^{\infty} \frac{(-1)^n}{(4n+1)!!}$$
Perhaps there isn't a closed form.  Double factorials are WAY out of my comfort zone so any tips would be appreciated. Thanks

Too long for a comment.
Using the same approach as in @Xoque55's answer, we could go one step further and consider
$$f(x)=\sum_{n=0}^{\infty} \frac{(-1)^n}{(4n+1)!!}x^{4n}=\, _1F_2\left(1;\frac{3}{4},\frac{5}{4};-\frac{x^4}{16}\right)$$ which write
$$f(x)=\frac{\sqrt \pi}x \left(C\left(\frac{x}{\sqrt{\pi }}\right) \cos
   \left(\frac{x^2}{2}\right)+S\left(\frac{x}{\sqrt{\pi }}\right) \sin
   \left(\frac{x^2}{2}\right) \right)$$
