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Old 2019-02-17, 05:57   #1
rudy235
 
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Jun 2015
Vallejo, CA/.

312 Posts
Default When does this series diverge?

assume the series
a ,aa, a^^aa 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 1.4222.05738816750076 ~ 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 e1/e Or the eth root of e =1.44466786100977

After about 190 iterations term a190 ~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 > e1/e WILL MAKE THE SERIES DIVERGENT?
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