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Is there a similar trivial equation for solving $\sum^h_{k=0}(m^k)$ like for solving $\sum^h_{k=0}(k) = \frac{n(n+1)}{2}$ as I want to evaluate the smallest h which when used will fulfill the equation $n \leq \sum^h_{k=0}(m^k)$

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up vote 3 down vote accepted

The sum $1+m+m^2+\cdots+m^h$ is a finite geometric series. It has (for $m\ne 1$) the sum $$\frac{m^{h+1}-1}{m-1}.$$

However, for your application, this is not directly useful, and estimates will be necessary. The strategy depends on the sizes of the numbers involved.

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