Determine convergence or divergence of the series.
-$$\sum _ { n = 2 } ^ { \infty } \frac { 6 } { n [ \ln ( 4 n ) ] ^ { 5 / 3 } }$$
A) Diverges
B) Converges

Question

Quizplus · Accepted Answer

The series $\sum _ { n = 2 } ^ { \infty } \frac { 6 } { n [ \ln ( 4 n ) ] ^ { 5 / 3 } }$ can be compared to the integral $\int _ { 2 } ^ { \infty } \frac { 6 } { x [ \ln ( 4 x ) ] ^ { 5 / 3 } } dx$ using the integral test for convergence. Since the function $f(x) = \frac { 6 } { x [ \ln ( 4 x ) ] ^ { 5 / 3 } }$ is positive, continuous, and decreasing for $x \geq 2$, the integral test can be applied. The integral converges because the exponent of the logarithm is greater than 1, which ensures that the integral of the function over $[2, \infty)$ has a finite value. Therefore, by the integral test, the series converges.

Determine Convergence or Divergence of the Series ∑n=2∞6n[ln⁡(4n)]5/3\sum _ { n = 2 } ^ { \infty } \frac { 6 } { n [ \ln ( 4 n ) ] ^ { 5 / 3 } }n=2∑∞​n[ln(4n)]5/36​

Determine Convergence or Divergence of the Series $\sum _ { n = 2 } ^ { \infty } \frac { 6 } { n [ \ln ( 4 n ) ] ^ { 5 / 3 } }$