TITLE: How we can find an equivalent of $\sum_{i=0}^n2^{2^i} $
QUESTION [5 upvotes]: How can we find an equivalent of the following sum:
$$\sum_{i=0}^n2^{2^i}$$
Because using integrals I have been recently able to determine equivalences for a lot of sums using Riemann's theorem (sum to intgeral) but I'm not able to tackle this one. Any help, suggestions?

REPLY [2 votes]: With Stolz-Cesaro, we have $\displaystyle\sum_{i=0}^n2^{2^i}\sim 2^{2^n}$.