Infinite Factorial Summation with Sum of the powers of 2

Calculus Level 4

S=1+1+22!+1+2+223!+1+2+22+234!+\large S= 1 + \dfrac{1+2}{2!} + \dfrac{1+2+2^2}{3!} + \dfrac{1+2+2^2+2^3}{4!} + \cdots

Given that S=Ae2+Be+CS = Ae^2 + Be + C , where A,BA,B and CC are integers, and e=limx0(1+x)1/x2.718\displaystyle e= \lim_{x\to 0} (1+x)^{1/x} \approx 2.718 , find the value of A×B×CA\times B \times C.


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