View Single Post
Old 2019-02-17, 15:12   #3
Dr Sardonicus
Dr Sardonicus's Avatar
Feb 2017

1101100111112 Posts

Originally Posted by rudy235 View Post
I have proven heuristically that the highest number of a for which the series still converges is e1/e ~ 1.44466786100977
The term an n--> ∞ . is e (2.718281828...)
Assuming the limit x satisfies

a^{x} \;=\;x

we have


It is an easy exercise to prove that the largest value of

y = x^{\frac{1}{x}}

for positive real x (logarithmic differentiation works nicely) is


This occurs at

x\; =\; e\text{.}

Last fiddled with by Dr Sardonicus on 2019-02-17 at 15:14 Reason: ginxif opsty
Dr Sardonicus is offline   Reply With Quote