Help with the Divergence Test

Question

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?

Answer 1

Let $k=e^x$, notice that when $x\to\infty$, then also $k\to\infty$.

Now, $$\lim_{k\to\infty}\frac{\sqrt{k}}{\ln^{10}k}=\lim_{x\to\infty}\frac{\sqrt{e^x}}{x^{10}}=\infty$$

Answer 2

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, $$\lim_{t\to \infty} \dfrac{e^{t/2}}{t^{10}} > \lim_{t\to \infty} \dfrac{t}{11! \cdot 2^{11}} = \infty$$