Re: x^(x^(x^(x^x^...)))....)))) = 2

if series doesn't blow up, then series will have a
limit point as it is monotone sequence.
So say series has limit y

I should not have used word series
as series has different meaning than sequence.
So please consider the word series in above
as sequence. (i.e. x_n = x^(x_{n-1}) x_0 = 1)

then x^y = y
or x = y^(1/y) ie. if at all the series converges,

series should be sequence

we should be able to write x as y^(1/y).

As Jonathan Hoyle posted,
(http://mathforum.org/kb/message.jspa?messageID=622068
3&tstart=0)
x is always between [o, 1.4446...]
Surely that doesn't mean for all x the series may

again series should be sequence


converge.

Then answer to your question would be <1.4446....
Still it is not answered as converse may not true.

But my doubt is how to differentiate function
y^(1/y)?
As I want to find maximum, I am have to find its
first derivative and second derivative.

I have plotted the graph for function y^(1/y)
using matlab. (granularity 0.0025, attached here in
y.jpg).
So, I just could check Jonathan Hoyle's statement by
matlab, but how should find the maximum in general
of
y^(1/y)? without help of computer?

---
Regards,
Sujit P Gujar.
IISc Bangalore.
Web : http://people.csa.iisc.ernet.in/sujit

---
Regards,
Sujit P Gujar.
IISc Bangalore.
Web : http://people.csa.iisc.ernet.in/sujit
.