The napierian logarithmic base e is also a special constant. The function y=ex is the natural exponential function and has interesting properties. It is the only function that is it's own derivative and it's own integral. IF: y=ex, then: y'=ex, y"=ex, etc. If x=1, y=e and the slope of y=ex is also e and the rate of change of slope at x=1 is also e. Who made it work that way? Also, the integral of exdx=ex. Kudos for creation.
The above summation very rapidly converges to the constant 'e' in 15 steps to 11 places, e=2.71828182846...
This function also converges to 'e' in one step as k approaches infinity. My laptop yields 2.7182818 for k=45E+6, math seems to run out of exponentiation accuracy for k.45E+6.
This function has a value approaching y=1 at x=0 and y=1 at x=1, with a minima in between. The function grows exponentially for x>1. This function has an interesting minima that exposes the base 'e'. The minima can be found using logarithmic differentiation to be at x=1/e. That is a minima at y=(1/e)(1/e)=0.692200628. Look at the form of the equation: y=xx, similar to y=ex, but charts a neat sort of 'parabolic looking' curve with a minima involving e located at (x,y) = (1/e,(1/e)(1/e)).
Is this not a surprising relationship?, tying (e), (PI), and (i) into a neat bundle where (i)=(-1)½
Maybe things like integer summations lead to a quotation attributed to mathematician Leopold Kronecker who once wrote that "God made the integers; all else is the work of man". And Einstein said "did God have a choice when creating the universe?" I think that question related to his wonder about the blemish free way that the fabric of the Universe hangs together. See, if one were sitting around designing a Universe, would one remember to tie all these functions and constants and summations together as they are?