# Closed form for the series $\sum^{\infty}_{x=2}\frac{1}{x!-1}$ and related ones

Does anyone know if there a closed-form expression for:

A. $\displaystyle\sum^{\infty}_{x=2}\frac{1}{x!-1}$

B. $\displaystyle\sum^{\infty}_{x=0}\frac{1}{x!+1}$

C. $\displaystyle\sum^{\infty}_{x=0}\frac{1}{x!+x}$

I guess there should be, but again, I couldn't come up with anything. Wolfram/Alpha did no help.

• Why do you think there should be closed forms for these series? – Yuriy S Apr 3 '18 at 19:19
• Mathematica can't find a closed form for any of this – Yuriy S Apr 3 '18 at 19:20
• If you come up with an expression for B then you have one for C (factorize the denominator and then use the generating function of the sequence in B). Having said that, I think the name of the summation variable is unfortunate – Tal-Botvinnik Apr 3 '18 at 20:30
• The digits of B are given here: http://oeis.org/A217702. No closed form is given. – Jair Taylor Apr 3 '18 at 20:35
• You may invent a special set of constants $$C_k = \sum_{n\geq 1}\frac{1}{n!^k}$$ ($C_1,C_2$ are already given by the exponential function and the Bessel function $I_0$) and define $A,B,C$ in terms of them. – Jack D'Aurizio Apr 3 '18 at 23:02