What is the value of p that marks the boundary between convergence and divergence of the series $\sum _ { n = 2 } ^ { \infty } \frac { 1 } { n ( \ln n ) ^ { y } }$ ?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 3 }$
C)Diverges for all p
D)ln 2
E)1
F)ln 3
G)Converges for all p
H) $\frac { 1 } { e }$

Question

What is the value of p that marks the boundary between convergence and divergence of the series $\sum _ { n = 2 } ^ { \infty } \frac { 1 } { n ( \ln n ) ^ { y } }$ ?
A) $\frac { 1 } { 2 }$ 
B) $\frac { 1 } { 3 }$ 
C)Diverges for all p
D)ln 2
E)1
F)ln 3
G)Converges for all p
H) $\frac { 1 } { e }$

Quizplus · Accepted Answer

The series $\sum _ { n = 2 } ^ { \infty } \frac { 1 } { n ( \ln n ) ^ { p } }$ is a p-series, and its convergence depends on the value of p. The series converges if $p > 1$ and diverges otherwise. The correct value of p that marks the boundary between convergence and divergence is 1, making choice E correct.

What Is the Value of P That Marks the Boundary ∑n=2∞1n(ln⁡n)y\sum _ { n = 2 } ^ { \infty } \frac { 1 } { n ( \ln n ) ^ { y } }∑n=2∞​n(lnn)y1​

What Is the Value of P That Marks the Boundary $\sum _ { n = 2 } ^ { \infty } \frac { 1 } { n ( \ln n ) ^ { y } }$