$\text { Find the positive values of } p \text { for which the series } \sum _ { n = 2 } ^ { \infty } \frac { 1 } { n ( \ln n ) ^ { p } } \text { converges. }$
A) converges for $p 0$
B) diverges only at $p = 1$
C) converges for $p \geq 2$
D) converges for $p 1$
E) converges for $p \geq 1$

Question

$\text { Find the positive values of } p \text { for which the series } \sum _ { n = 2 } ^ { \infty } \frac { 1 } { n ( \ln n ) ^ { p } } \text { converges. }$ 
A) converges for $p > 0$
B) diverges only at $p = 1$
C) converges for $p \geq 2$
D) converges for $p > 1$
E) converges for $p \geq 1$

Quizplus · Accepted Answer

The series converges for $p > 1$ based on the integral test for convergence. For $p \leq 1$, the integral of $1 / [n (\ln n)^p]$ diverges, indicating that the series also diverges. For $p > 1$, the integral converges, implying that the series converges.

A) Converges For p>0p > 0p>0 B) Diverges Only At p=1p = 1p=1

A) Converges For $p > 0$ B) Diverges Only At $p = 1$