TITLE: Exponential on itself series converges
QUESTION [0 upvotes]: My first approach is that I need to use induction on that. However the problem is that what is the upper bound limit for this series?
Also, let's say we showed that it converges. Doesn't it directly imply that that convergence point is the limit of the series? What is the difference?
Note: This is non-grading homework question. Since it is explained in image, I feel that I need to explain.

REPLY [0 votes]: Once you know that the sequence converges, what can you infer about the actual value of $a:=\lim a_n$?
Since $f\colon x\mapsto \sqrt2^x$ is continuous on $\Bbb R$, we must have $$\tag1a=\sqrt 2^a.$$ As $f$ is also strictly increasing, $(1)$ cannot have more than one solution. One (and hence the) solution we can immediately spot is $$a=2.$$
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