For my infinite limits to work perfectly, I need to define a “smooth” function.
“Smooth” excludes functions with periodic, random and chaotic components.
For a while I thought that if a Taylor series had monotonically decreasing positive coefficients then that sufficed to define “smooth”.
Not so.
Define
A = 1 + x^4/4! + x^8/8! + x^12/12! + x^16/16! + etc
B = x + x^5/5! + x^9/9! + x^13/13! + x^17/17! + etc
C = x^2/2 + x^6/6! + x^10/10! + x^14/14! + x^18/18! + etc
D = x^3/3! + x^7/7! + x^11/11! + x^15/15! + x^19/19! + etc
A+B+C+D = exp(x) is smooth
A-C = cos(x) is pure periodic, not smooth
B-D = sin(x) is pure periodic, not smooth
A-B+C-D = exp(-x) is smooth despite negative Taylor series coefficients
A+C = cosh(x) is smooth
B+D = sinh(x) is smooth
A = (cosh(x)+cos(x))/2
B = (sinh(x)+sin(x))/2
C = (cosh(x)-cos(x))/2
D = (sinh(x)-sin(x))/2
So, A, B, C & D all have periodic fluctuating components so are not “smooth”.
Despite all having positive, monotonically decreasing, Taylor series coefficients.
Damn.