Determine whether the series is conditionally convergent, absolutely convergent, or divergent.
$$\sum_{n=2}^{\infty}\frac{(-1)^n\sqrt{n}}{\ln(n)}$$
The absolute value of this sum is divergent by the divergence test, and it's inconclusive by the ratio and root tests, and the alternating series test doesn't apply because the sequence isn't decreasing. I think all of this only tells me that my conclusions are inconclusive. Is this correct? What can I do to actually determine if this series is absolutely convergent, conditionally convergent, or divergent?
Best Answer
Note that $$ \lim_{n\to\infty}\frac{\sqrt{n}}{\ln(n)}\stackrel{L}{=}\lim_{n\to\infty}\frac{\frac{1}{2\sqrt{n}}}{\frac{1}{n}} =\frac{1}{2}\lim_{n\to\infty}\frac{n}{\sqrt{n}} =\frac{1}{2}\lim_{n\to\infty}\sqrt{n}=\infty $$ This implies that your sum diverges.