Does $x^2+ x^4/(3\cdot4) + x^6/(3\cdot4\cdot5\cdot6) + \cdots$ have any compact form? Is there any compact form for the following series
$$F_1(x) = x^2+ \frac{x^4}{3\cdot4} + \frac{x^6}{3\cdot4\cdot5\cdot6} + \cdots$$
$$F_2(x) = x+ \frac{x^3}{2\cdot3} + \frac{x^5}{2\cdot3\cdot4\cdot5} + \cdots$$
$$F_3(x) = F_2(x) - F_1(x)$$

Yes,
$F_1(x) = 2(\cosh x - 1)$;
$F_2(x) = \sinh x$.
(see Taylor series).
