When does this series diverge?
assume the series
a ,a^{a}, a^^a^{a} and succesively...
For a=1 for instance, all terms are 1
thus the series is 1, 1, 1, .....1.1 it does not diverge.
For a= SQRT(2) = 1.41421356..
the series is 1.41421356, 1.6325269, 76083956 ... and at approximately term 57 it converges to 2.00000000
This would be a great place to stop.
However if a=1.42 it still seems to converge. At term 74 it seems to stop growing at 2.05738816750076 in other words 1.422^{2.05738816750076} ~ 2.05738816750076
I tried 1+4/9 =1.4444...
This one takes longer but around term 135 it seems to stabilize at 2.63947300401328
My next try was e^{1/e} Or the eth root of e =1.44466786100977
After about 190 iterations term a_{190 } ~2.69004748029863
I believe this is the absolute limit for the series to converge. A slightly bigger number 1.445 diverges rather quickly.
IS THERE SOME ANALYTICAL PROOF THAT ANY NUMBER > e^{1/e} WILL MAKE THE SERIES DIVERGENT?
