Define $$\phi(a,b,k) = e^{-a/b} + e^{-a/(b+1)} + \cdots + e^{-a/(b+k)}.$$ For example, $$\phi(10,2,3) = e^{-10/2} + e^{-10/3} + e^{-10/4} + e^{-10/5}, $$ or $$\phi(10,2,3) = e^{-5} + e^{-3.33..} + e^{-2.25} + e^{-2}.$$ We will assume that $a,b$ are positive integers, $a>b$, and $k$ is an integer.
I need to upper bound this. Intuitively, it seems clear that only the last terms of the sum really matter, so something like $$\phi(a,b,k) \approx e^{-a/(b+k)}$$ should hold where $\approx$ perhaps hides some moderately growing function. What is a formal statement that captures this?
How about noting that since $x\mapsto e^{-x}$ is a decreasing function, you have that $e^{-a/(b+j)}\leq e^{-a/(b+k)}$ for all $k\geq j\geq 0$, and then bound your function naively by
\begin{align*} \phi(a,b,k) &= e^{-a/b} + e^{-a/(b+1)} + \cdots + e^{-a/(b+k)}\\ &\leq e^{-a/(b+k)}+e^{-a/(b+k)}+\cdots+e^{-a/(b+k)}\\ &= (k+1)e^{-a/(b+k)}. \end{align*}