Determine all values of c such that the series converges.
A) any real number
B) c in the interval
C) c in the interval
D) c in the interval

Question

Determine all values of c such that the series converges. 
A) any real number 
B) c in the interval 
C) c in the interval 
D) c in the interval

Quizplus · Accepted Answer

The series $\sum_{n=1}^\infty \frac{1}{n(\ln n)^c}$ is a form of the well-known p-series $\sum_{n=1}^\infty \frac{1}{n^p}$, where $p=c-1$ and $p>0$. We know that the p-series converges if $p>1$ and diverges if $p\leq 1$. Thus, we have $c-1>1$, or $c>2$. Therefore, the values of $c$ such that the series converges must lie in the interval $(2,\infty)$, which is option (C).

Determine All Values of C Such That the Series Converges