kayak

Let $m(n)=\min\{m:2^{-m}m^{-3/2}\leq n^{-1}\}$. Then $$ 2^{m(n)}\sim\frac{n}{\log_{2}n}. $$ This problem is a piece of solving : An unfair "fair game."

This is problem 2.2.8 from Durrett's Probability Theory and Examples 4th edition, I am using the version of this book that can be found on his website. Let $p_k=\frac{1}{2^k k (k+1)}, \ k=1,2,\dots$ and $p_0=1-\sum_{k\geq 1}p_k.$ $$ \sum\limits_{k=1}^\infty 2^k p_k = \left(1-\frac{1}{2}\rig...