Let $p$ and $q$ be positive real numbers such that $q < p$ , then is the following series convergent? $$ \sum\limits_{n=2}^\infty(-1)^n\frac{(\ln n)^p}{n^q} $$
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The series converges by the alternating series test. The $n^{\text{th}}$ term $a_n = \dfrac{(\ln n)^p}{n^q}$ is the term of a decreasing to $0$ sequence. To show this is true, consider the function: $f(x) = \dfrac{(\ln x)^p}{x^q}$ on $[2, \infty)$. The derivative $f'(x)$ is negative if and only if $e^{p/q} < x$. This means the sequence $\{a_n\}$ decreases when $n > e^{p/q}$ hence the conclusion.
