A question from Introduction to Analysis by Arthur Mattuck:
Let $n!!=n(n-2)(n-4)\cdot…\cdot k$, where $k=1$ or $2$,depending on whether n is odd or even. (define $0!!=1$.)
Evaluate the sum $f(x)=\sum_0^\infty\frac{x^n}{n!!},$ using term-by-term differentiation and integration.
I think what the question asked is to give an explicit form for this sum.
If you differentiate,
$$f'(x)=\sum_{k=1}^\infty\frac{nx^{n-1}}{n!!}=1+\sum_{k=2}^\infty\frac{x^{n-1}}{(n-2)!!}=1+xf(x).$$
This is a linear differential equation. The homogeneous part is separable and yields
$$f'(x)=xf(x)\to f(x)=Ke^{x^2/2}.$$
Then by variation of the constant,
$$K(x)=\int e^{-x^2/2}dx+C,$$
$$f(x)=e^{x^2/2}\int e^{-x^2/2}dx+Ce^{x^2/2}.$$
With the initial condition $f(0)=1$,
$$f(x)=e^{x^2/2}\int_{t=0}^x e^{-t^2/2}dt+e^{x^2/2}.$$