Does the following limit exist ?

limn→∞exp(−1+exp(−2+exp(−3+⋯exp(−n)⋯)))

If yes, can it be expressed in a closed form ?

PARI shows the following numerical value :

`n=-100;x=exp(n);while(n<-1,n=n+1;x=exp(x+n));print(x)`

0.4241685586940448516119410516

Within this precision, −200 yields the same value.

**Answer**

The limit exists because the sequence is monotonic and bounded, although I don’t know yet what the limit is. Let’s denote this sequence En.

The sequence is monotonic because −n+exp(−(n+1))>−n, therefore exp(−n)+exp(−(n+1)))>exp(−n), therefore exp(−(n−1)+exp(−n)+exp(−(n+1))))>exp(−(n−1)+exp(−n)), etc, until you get En+1>En.

The sequence is bounded because −n+exp(negative number)<0, therefore En<exp(0)=1.

By monotone convergence theorem En converges.

**Attribution***Source : Link , Question Author : Peter , Answer Author : Michael*