I am interested in calculating the following series
$$ \sum_{k=0}^{+\infty}{ \frac{ x^k }{\, a (a+1) \cdots (a+k) \,} } $$
using this continued fraction (I expect) equivalent form:
$$ \cfrac{1}{a + \cfrac{-ax}{a+1 + \cfrac{x}{a+2 + \cfrac{-(a+1)x}{a+3 + \cfrac{2x}{a+4 + \cfrac{-(a+2)x}{a+5 + \cfrac{3x}{a+6 + \cdots}}}}}}} \, \mathrm{,} $$
which can be expressed as
$$ \cfrac{f_0}{g_0 + \cfrac{f_1}{g_1 + \cfrac{f_2}{g_2 + \cdots}}} \, \mathrm{,} $$
where
$$ f_k = \left\{ \begin{aligned} 1, &&& \text{if $k=0$}\\ \frac{k}{2}x , &&& \text{if $k$ is even, $k \neq 0$} \\ -\left(a+\frac{k-1}{2} \right)x , &&& \text{if $k$ is odd} \end{aligned} \right. $$
[NOTE: The definition of $f_k$ was initially wrong, and has been fixed according to one of the answers.]
and
$$ g_k = a + k \, \text{,} $$
for $k \geq 0$.
I have tried to implement this continued fraction in Mathematica. This is what I have done:
f[k_, a_, x_] := If[EvenQ[k], (k/2)*x, -(a + (k - 1)/2)*x];
g[k_, a_] := a + k;
CONTFRAC[a_, x_] :=
ContinuedFractionK[f[k, a, x] // Evaluate,
g[k, a] // Evaluate, {k, 0, +Infinity}];
Then I try
N[CONTFRAC[3/2, -100/3]]
(which should return something similar to $0.029542920686933995848485613587409381469661178405257$), but all I get is this:
So how can/should I ask Mathematica to calculate this continued sum?
NOTE: Why am I interested in doing that? Mathematica numerically calculates the value of the series with enough accuracy, but in other programming environments it is not like that for some values of $a$ and $x$ (for more details, see this math.SE thread).