Show that if c>l, then มะ ) Converges
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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Question
![**Mathematical Convergence Problem**
**Objective:** Demonstrate that for \( c > 1 \), the series
\[
\sum_{n=1}^{\infty} \frac{1}{n (\ln n)^c}
\]
converges.
**Explanation:**
This problem involves analyzing the convergence of an infinite series with the general term
\[
a_n = \frac{1}{n (\ln n)^c}
\]
where \( \ln \) denotes the natural logarithm and \( c \) is a constant greater than 1. The series starts from \( n = 2 \) (since \( \ln 1 = 0 \) would lead to division by zero at \( n=1 \)) and continues indefinitely.
**Approach:**
To show the convergence, a common method is to use the **Integral Test** or to compare it to known convergent series. The integral test states that if \( f(n) = \frac{1}{n (\ln n)^c} \) is continuous, positive, and decreasing, then the convergence of the integral
\[
\int_{2}^{\infty} \frac{1}{x (\ln x)^c} \, dx
\]
will determine the convergence of the series.
For this problem, analyze the integral by substitution, given the condition \( c > 1 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe57a7d70-87de-4a1f-8104-5b2578062c6c%2F8747afa5-398e-4354-80de-eef529054038%2Fjiqgwqj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Mathematical Convergence Problem**
**Objective:** Demonstrate that for \( c > 1 \), the series
\[
\sum_{n=1}^{\infty} \frac{1}{n (\ln n)^c}
\]
converges.
**Explanation:**
This problem involves analyzing the convergence of an infinite series with the general term
\[
a_n = \frac{1}{n (\ln n)^c}
\]
where \( \ln \) denotes the natural logarithm and \( c \) is a constant greater than 1. The series starts from \( n = 2 \) (since \( \ln 1 = 0 \) would lead to division by zero at \( n=1 \)) and continues indefinitely.
**Approach:**
To show the convergence, a common method is to use the **Integral Test** or to compare it to known convergent series. The integral test states that if \( f(n) = \frac{1}{n (\ln n)^c} \) is continuous, positive, and decreasing, then the convergence of the integral
\[
\int_{2}^{\infty} \frac{1}{x (\ln x)^c} \, dx
\]
will determine the convergence of the series.
For this problem, analyze the integral by substitution, given the condition \( c > 1 \).
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