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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### 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.
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### 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.
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