EDIT: I figured out a solution; see my comment below.
denotes the exponential integral;
denotes the Euler-Mascheroni constant, .
Notation: denotes the harmonic number, .
I came across this limit when I was trying to derive the derivative of the factorial function: for all whole numbers . Let's denote the derivative as We know that . Therefore, differentiating gives us:
It follows that for some constant . What I can't figure out is how to prove that . I tried to find the value of :
Using integration by parts, this becomes:
We can simplify this a little by observing the following:
Hence, the equation for can simplify to:
Now this is supposed to equal , but I cannot figure out how to prove it, and can't seem to find the proof anywhere either.