How to understand the probablistic analysis of skiplist?
The first sum is obviously equal to $\lfloor\log n\rfloor\leq\log n$.

For the second sum, assuming that logs are base-$2$ and using $r&gt;\lfloor\log n\rfloor$,

$$2^r = 2^{\lfloor\log n\rfloor+1}2^{r-\lfloor\log n\rfloor-1} \geq n2^{r-\lfloor\log n\rfloor-1}\,,$$

and the result follows.  If the logs aren't base-$2$, then there'll be a scaling factor but the second sum will still be $\Theta(1)$.