Searching for an asymptotic to exp[0.5]

[tommy1729]

One obvious questions from the Taylor series result, that I can't answer, because I have no idea how fast these functions grow as x goes to infinity, relative to exponentiaton. What is the "growth" of a functions like these, which should grow slower than exponentiation, but faster than any polynomial?
\( \sum_{n = 1}^{\infty}\frac{x^n}{(2n)!} \)
\( \sum_{n = 1}^{\infty}\frac{x^n}{(4n)!} \)

I mentioned these before.

They are the " fake " exp(sqrt(x)) and exp(sqrt(sqrt(x))).

since a sqrt(x) is much closer to x than any positive iterate of a logaritm it follows that they both also have growth = 1.

These sums are related to the linear ordinary differential equations.
The first is cosh(sqrt(x)).
cosh(ln(x)) < 2cosh(sqrt(x)) < cosh(x).
SO growth = 1 follows.

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Also like I said before the 0 terms do not change that much.
Maybe that makes more sense now.
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Thanks for the data !


regards

tommy1729