formulate a power series

Question

Could anyone advise me if there is any way to estimate/formulate the following series

$$ \sum_{i=m..n} \left(\frac{c}{i}\right)^i $$

where m,n and c are positive integers.

Answer 1

There may be a better estimate, but I can get an upper bound to be $e^c$ as follows $$ \sum_{i=m}^n\dfrac{c^i}{i^i}\leq \sum_{i=m}^n\dfrac{c^i}{i!}\leq e^c.$$

Of course this is assuming you mean the sum from $m$ to $n$, and this upper bound becomes a very bad estimate as $m$ grows.