P 1. Use the integral test to determine whether the series En-2 n(In(n))3/4 convergent or divergent. is
P 1. Use the integral test to determine whether the series En-2 n(In(n))3/4 convergent or divergent. is
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.3: Geometric Sequences
Problem 49E
Related questions
Question
![---
### P 1.
Use the integral test to determine whether the series
\[ \sum_{n=2}^{\infty} \frac{1}{n (\ln(n))^{3/4}} \]
is convergent or divergent.
### Explanation
The task involves applying the integral test to analyze the series. According to the integral test, if \( f(x) \) is a continuous, positive, and decreasing function for \( x \geq a \), and \( a_n = f(n) \), then the series \( \sum_{n=a}^{\infty} a_n \) converges if and only if the improper integral \( \int_{a}^{\infty} f(x)\,dx \) converges.
In this given example:
\[ a_n = \frac{1}{n (\ln(n))^{3/4}} \]
we need to set up and evaluate the corresponding integral:
\[ \int_{2}^{\infty} \frac{1}{x (\ln(x))^{3/4}}\,dx \]
Calculating and analyzing the convergence of this integral will determine whether the series converges or diverges.
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa216c9ee-ecac-40ec-b583-a361985d4bbd%2F3fc0fcdb-7bb0-468c-91d4-f5c87063ae62%2F28xhvac_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
### P 1.
Use the integral test to determine whether the series
\[ \sum_{n=2}^{\infty} \frac{1}{n (\ln(n))^{3/4}} \]
is convergent or divergent.
### Explanation
The task involves applying the integral test to analyze the series. According to the integral test, if \( f(x) \) is a continuous, positive, and decreasing function for \( x \geq a \), and \( a_n = f(n) \), then the series \( \sum_{n=a}^{\infty} a_n \) converges if and only if the improper integral \( \int_{a}^{\infty} f(x)\,dx \) converges.
In this given example:
\[ a_n = \frac{1}{n (\ln(n))^{3/4}} \]
we need to set up and evaluate the corresponding integral:
\[ \int_{2}^{\infty} \frac{1}{x (\ln(x))^{3/4}}\,dx \]
Calculating and analyzing the convergence of this integral will determine whether the series converges or diverges.
---
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage