TITLE: Help with the Divergence Test
QUESTION [0 upvotes]: I have the series
\begin{equation}
\displaystyle\sum_{k=2}^\infty\frac{\sqrt{k}}{\ln^{10}k}
\end{equation}
and I want to use the Divergence Test to determine whether it diverges or not. Do I really need to use L'Hôpital's rule 10 times to prove that it diverges?

REPLY [1 votes]: Note that for a series to converge, the $n^{th}$ term has to go to zero. However, if we consider the $n^{th}$ term of the series, we have $a_n = \dfrac{\sqrt{n}}{\ln^{10}(n)}$. And,
$$\underbrace{\lim_{n\to \infty} \dfrac{\sqrt{n}}{\ln^{10}(n)} = \lim_{t\to \infty} \dfrac{e^{t/2}}{t^{10}}}_{n=e^t}$$
Now note that
$$e^{t/2} = \sum_{k=0}^{\infty} \dfrac{(t/2)^{k}}{k!} > \dfrac{(t/2)^{11}}{11!}$$
Hence,
$$\lim_{t\to \infty} \dfrac{e^{t/2}}{t^{10}} > \lim_{t\to \infty} \dfrac{t}{11! \cdot 2^{11}} = \infty$$
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