TITLE: Closed form formula for $2^{2^1}+2^{2^2}+...+2^{2^n}$
QUESTION [5 upvotes]: So, the series's sum is of below form:
$2^{2^1}+2^{2^2}+...+2^{2^n}$
This series is an intermediate work of an bigger problem {So I am concerned with whether $2^{2^n}-\frac{\sum_{i=1}^n(2^{2^i})}{2}$ is anywhere close to $2^{2^{n-1}}$}. With limited knowledge of series, I am stuck.
Thank you.

REPLY [3 votes]: The simple thing is to look at the ratio of the two largest terms.  $$\frac {2^{2^n}}{2^{2^{(n-1)}}}=2^{2^n-2^{(n-1)}}=2^{2^{(n-1)}}$$ which grows rapidly with $n$.  The smaller terms are really negligible and the sum is very close to $2^{2^n}$.  Note that you should be adding to, not subtracting from, the largest term.  My argument says you don't have to bother.