Take this partial sum. By the formula for a finite geometric series:
$$ \sum_{k=1}^n2\pi^{k-1}=2\left(\frac{1-\pi^n}{1-r}\right) $$
Simple enough. However, when a $\operatorname{ceil}$ is introduced, this becomes far less trivial:
$$ \sum_{k=1}^n\left\lceil2\pi^{k-1}\right\rceil $$
Is there a method by which this can be simplified such that it does not necessitate summation? At the very least, could the value be computed in a more efficient manner than directly summing the terms?