Formula or asymptotic behavior of a partial sum

Question

I'm wondering if there is a known formula for the partial sum given by $$ \sum_{k=1}^n e^{\sqrt{k}} $$ If not, could someone explain how one might deduce the asymptotic behavior of this sum? For example, is it rate equivalent to the sequence $n e^{ \sqrt{n} }$ ?

Answer 1

Comparing with integrals: $$\int_0^{n}e^{\sqrt{u}}du\leq\sum_{k=1}^ne^{\sqrt{k}}\leq\int_1^{n+1}e^{\sqrt{u}}du\leq\int_0^{n+1}e^{\sqrt{u}}du$$ Now, $$\int_0^{n}e^{\sqrt{u}}du=\int_0^{\sqrt{n}}e^{x}2xdx=\sqrt{n}e^{\sqrt{n}}-e^{\sqrt{n}}+1 \text{ (integration by parts)}$$ Thus:$$\boxed{\sum_{k=1}^ne^{\sqrt{k}}\simeq\sqrt{n}e^{\sqrt{n}}}$$