# Closed-form of infinite continued fraction involving factorials

Is there a closed form of this:

$$1!+\dfrac{1}{2!+\dfrac{1}{3!+\dfrac{1}{4!+\ldots}}}$$

• It is interesting . and can't see it.hope someone take some usefull [Continued fraction]:mathworld.wolfram.com/ContinuedFraction Dec 19, 2014 at 11:36
• not too far from $459/314$ Dec 19, 2014 at 11:47
• $1.461783355000579602560079367$ is the value of the number. Dec 19, 2014 at 12:17
• $\large \frac{55099}{37693}$ is an approximation with an error of abot $10^{-12}$ Dec 19, 2014 at 12:37
• @ajotatxe Careful with that bet or you might end up like Robespierre! (I guess "almost all" real numbers are transcendental, so you might be safe ...) Dec 19, 2014 at 17:00

My 2 cents worth. Here is a Pascal program snippet that does Peter's job; actually half of it due to Delphi's double precision limitations. Backward recursion is the clue (again).

program Peter;
procedure fraction;
var
a : double;
f,k : integer;
begin
f := 1*2*3*4*5*6*7*8*9*10;
k := 10; a := f;
while k > 1 do
begin
f := f div k;
a := 1/a+f;
k := k-1;
end;
Writeln(a);
end;
begin
fraction;
end.

Output:
 1.46178335500058E+0000

Closed form? I don't think so.