Can $\sum_{n=2}^\infty \frac1{n(\ln n)^p}$ be treated as p-series

Question

The question is to find out whether the following series converges: $$\sum_{n=2}^\infty \dfrac1{n(\ln n)^p}$$

Can this be treated as a p-series with $n\ln n$ as $x$ so it would be of the form $\dfrac1{x^p}$?

If it can't, it would help my understanding to see the reasoning, preferably with a counterexample explaining why. Thanks

Answer 1

The answer depends on the meaning of "treated as $p$-series".

• If the meaning is "be solved by the same tool (namely, integral test)" or "be convergent for the same values of $p$ (namely, $p>1$)", then the answer is yes.

• If the meaning is "to apply the same conclusions (without further analysis) because of algebraic similarity (namely, denominator going to infinity raised to the power $p$), then the answer is no. Counterexample:

$$\sum_{n=2}^\infty \frac{1}{\ln n (\ln n)^p}$$