In this Wikipedia article on a Non-analytic smooth function an estimate is given which is crucial to the argument that the Taylor series only represents the smooth function at the center.
The estimate which currently vexes me is: $$ \sum_{k \in A, k >q } e^{-\sqrt{k}}k^n \geq e^{-\sqrt{n}}n^n $$ as $n \rightarrow \infty$. In the above $q = 2^m$ for some fixed positive integer $m$ and $A = \{ 2^k \ | \ k \in \mathbb N \}$. My question is simply:
How can we justify the inequality above ?
I will award a 100pt bounty to whoever helps best as soon as it is allowed by the software of the site. Thanks in advance for your insight on this pesky inequality.
The article gives that $n \in A$, $n>q$ in the conditions used in deriving that inequality. But this just reduces the problem to a sum of positive terms being larger than one of the terms (i.e. if $a_k \geq 0$ for every $k \in K$, $\sum_{k \in K} a_k \geq a_n$ for any $n \in K$).