Answer:
Diverges
Step-by-step explanation:
since the value of P = 5/6 < 1 the p-series diverges
summation of 1/n^5/6 at n = 1 and infinity
attached below is the detailed solution using Theorem 9.11 to determine the convergence or divergence of the p-series
The value of the p in the p-series is less than one. Then the p-series is a divergence.
It is an infinite type of series.
The p-series is given below.
[tex]1 + \dfrac{1}{\sqrt[6]{32} } +\dfrac{1}{\sqrt[6]{243} }+\dfrac{1}{\sqrt[6]{1024} }+\dfrac{1}{\sqrt[6]{3125} }[/tex]
The series can be written as
[tex]\dfrac{1}{1^{5/6}}+\dfrac{1}{2^{5/6}}+\dfrac{1}{3^{5/6}}+\dfrac{1}{4^{5/6}}+\dfrac{1}{5^{5/6}}[/tex]
Then the sum will be
[tex]\rm \Rightarrow \Sigma _{n=0} ^{\infty} \dfrac{1}{n^{5/6}} \\\\\\\Rightarrow P = \dfrac{5}{6} < 1[/tex]
Since the value of the p is <1. Then the series is divergence.
More about the p-series link is given below.
https://brainly.com/question/24249275
