21. 4 m=1m! +4m
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
use the Direct Comparison Test to determine whether the infinite series is convergent.
![**21. \[ \sum_{m=1}^{\infty} \frac{4}{m! + 4^{m}} \]**
### Explanation:
This mathematical expression represents an infinite series where \( m \) starts at 1 and goes to infinity.
- **Summation Notation (\(\Sigma\)):** This symbol indicates that the terms of a sequence (in this case, \(\frac{4}{m! + 4^m}\)) are to be summed.
- **Starting Index \( m=1 \):** The summation begins at \( m = 1 \).
- **Infinity (\(\infty\)):** The summation continues indefinitely.
- **Term of the Series (\(\frac{4}{m! + 4^m}\)):** Each term in the series is a fraction. The numerator is 4, and the denominator is \( m! + 4^m \), where \( m! \) (m factorial) is the product of all positive integers up to \( m \), and \( 4^m \) is 4 raised to the power of \( m \).
### Usage Example on an Educational Website:
In this example, students would analyze the behavior of the given infinite series. They might be asked to determine the convergence or divergence of the series or to perform numerical approximations for the partial sums of the series. Understanding this summation enhances comprehension of series and sequences in calculus and mathematical analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F37708386-d0d8-46c3-a2d8-8e337c754037%2Fcc8afbfb-f110-4c07-91f8-e37183013f78%2Flmduj0n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**21. \[ \sum_{m=1}^{\infty} \frac{4}{m! + 4^{m}} \]**
### Explanation:
This mathematical expression represents an infinite series where \( m \) starts at 1 and goes to infinity.
- **Summation Notation (\(\Sigma\)):** This symbol indicates that the terms of a sequence (in this case, \(\frac{4}{m! + 4^m}\)) are to be summed.
- **Starting Index \( m=1 \):** The summation begins at \( m = 1 \).
- **Infinity (\(\infty\)):** The summation continues indefinitely.
- **Term of the Series (\(\frac{4}{m! + 4^m}\)):** Each term in the series is a fraction. The numerator is 4, and the denominator is \( m! + 4^m \), where \( m! \) (m factorial) is the product of all positive integers up to \( m \), and \( 4^m \) is 4 raised to the power of \( m \).
### Usage Example on an Educational Website:
In this example, students would analyze the behavior of the given infinite series. They might be asked to determine the convergence or divergence of the series or to perform numerical approximations for the partial sums of the series. Understanding this summation enhances comprehension of series and sequences in calculus and mathematical analysis.
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 3 steps with 3 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
