M n=1 n? + ln(n) 2n3 – 1
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
Determine whether each of the following series converges or diverges.
![The mathematical expression shown in the image represents an infinite series. The series is written in summation notation and is described as follows:
\[ \sum_{n=1}^{\infty} \frac{n^2 + \ln(n)}{2n^3 - 1} \]
Where:
- The symbol \(\sum\) denotes the summation operation.
- The limits of summation are from \(n = 1\) to \(n = \infty\).
- The expression inside the summation \(\frac{n^2 + \ln(n)}{2n^3 - 1}\) is the term that is being summed over all positive integer values of \(n\).
Explanation of Terms:
- \(n^2\) is the square of \(n\).
- \(\ln(n)\) is the natural logarithm of \(n\).
- The denominator \(2n^3 - 1\) is a polynomial in \(n\).
Understanding and analyzing this series involves determining whether the series converges or diverges, which requires knowledge of various convergence tests (such as the comparison test, ratio test, or integral test) used in the study of infinite series in mathematical analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd04de2b6-c3ee-4b26-bb77-15432c184cae%2F434b1481-5006-46d5-9fe1-d4b2ec05a5b9%2Fedvd0ln_processed.png&w=3840&q=75)
Transcribed Image Text:The mathematical expression shown in the image represents an infinite series. The series is written in summation notation and is described as follows:
\[ \sum_{n=1}^{\infty} \frac{n^2 + \ln(n)}{2n^3 - 1} \]
Where:
- The symbol \(\sum\) denotes the summation operation.
- The limits of summation are from \(n = 1\) to \(n = \infty\).
- The expression inside the summation \(\frac{n^2 + \ln(n)}{2n^3 - 1}\) is the term that is being summed over all positive integer values of \(n\).
Explanation of Terms:
- \(n^2\) is the square of \(n\).
- \(\ln(n)\) is the natural logarithm of \(n\).
- The denominator \(2n^3 - 1\) is a polynomial in \(n\).
Understanding and analyzing this series involves determining whether the series converges or diverges, which requires knowledge of various convergence tests (such as the comparison test, ratio test, or integral test) used in the study of infinite series in mathematical analysis.
![The image contains a mathematical expression representing an infinite series. The expression is as follows:
\[ \sum_{n=1}^{\infty} \frac{n \cdot 2^{3n}}{5^n} \]
Here's a detailed breakdown of the expression:
- The symbol \( \sum \) denotes a summation.
- The lower limit of the summation is \( n = 1 \), indicating that the summation starts from \( n = 1 \).
- The upper limit of the summation is \( \infty \), indicating that the summation continues infinitely.
- Inside the summation, the term being summed is \( \frac{n \cdot 2^{3n}}{5^n} \).
- \( n \) is a variable representing the index of summation.
- \( 2^{3n} \) represents an exponential function where the base is 2 and the exponent is \( 3n \).
- \( 5^n \) represents an exponential function where the base is 5 and the exponent is \( n \).
In summary, the expression is the infinite summation of the term \( \frac{n \cdot 2^{3n}}{5^n} \) starting from \( n = 1 \). This type of summation is commonly encountered in series calculus and discrete mathematics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd04de2b6-c3ee-4b26-bb77-15432c184cae%2F434b1481-5006-46d5-9fe1-d4b2ec05a5b9%2Frv58enk.png&w=3840&q=75)
Transcribed Image Text:The image contains a mathematical expression representing an infinite series. The expression is as follows:
\[ \sum_{n=1}^{\infty} \frac{n \cdot 2^{3n}}{5^n} \]
Here's a detailed breakdown of the expression:
- The symbol \( \sum \) denotes a summation.
- The lower limit of the summation is \( n = 1 \), indicating that the summation starts from \( n = 1 \).
- The upper limit of the summation is \( \infty \), indicating that the summation continues infinitely.
- Inside the summation, the term being summed is \( \frac{n \cdot 2^{3n}}{5^n} \).
- \( n \) is a variable representing the index of summation.
- \( 2^{3n} \) represents an exponential function where the base is 2 and the exponent is \( 3n \).
- \( 5^n \) represents an exponential function where the base is 5 and the exponent is \( n \).
In summary, the expression is the infinite summation of the term \( \frac{n \cdot 2^{3n}}{5^n} \) starting from \( n = 1 \). This type of summation is commonly encountered in series calculus and discrete mathematics.
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