Searching for an asymptotic to exp[0.5]

[tommy1729]

Fake function theory has been used mainly for non-entire functions.

But it can also be used for entire functions with some negative derivatives.
(as mentioned in the previous post)

And it can even be used to estimate entire functions that have all derivatives positive.

For those reasons , it seems likely that fake function theory will also have applications in number theory.

Lets take the simplest case : exp(x).

Using sheldon's post 9 (S9)

a_n x^n = exp(x)

ln(a_n) + n ln(x) = x

ln(a_n) = min ( x - n ln(x) )

d/dx [ x - n ln(x) ] = 1 - n/x

1 - n/x = 0

x = n

min ( x - n ln(x) ) = n - n ln(n)

ln(a_n) = n - n ln(n)

a_n = e^n/n^n

This approximates 1/n!

Although not so good.

My Q9 gives a different result.

Further investigation is needed.

Error term theorems are wanted.

regards

tommy1729