I was just pondering how will this series converge. Assuming $n \ge 0$, find $$ f(n)=n^{1/2} + n^{1/4} + n^{1/8} + n^{1/16} +\text{...}+ 1 $$ We stop the summation when we hit $1$ i.e. $\lfloor{n^{1/2^k}}\rfloor=1$
I tested how quickly the number converges i.e. how many items until its 1. It seems like it converges pretty quickly and the total sum is $O(\sqrt n)$ since I verified for the number $10^{10}$ and the computer returned $100341$(summing after taking the floor of the intermediate values).
It may be pretty easy problem but can't seem to find a way to derive a formula out of this. Any insights would be great. Thank you.