Use ratio test to determine if the series is absolutely convergent or not.
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 ratio test to determine if the series is absolutely convergent or not.
![The image presents the mathematical expression for an infinite series:
\[
\sum_{n=0}^{\infty} \frac{(-1)^n (1+i)^{2n}}{(2n)!}
\]
### Components of the Expression:
- **Sigma Notation (\(\sum\))**: Represents the summation of terms from \(n = 0\) to infinity (\(\infty\)).
- **\((-1)^n\)**: Alternating sign factor, which changes the sign of each term depending on whether \(n\) is even or odd.
- **\((1+i)^{2n}\)**: The base \(1+i\) is raised to the power of \(2n\). Here, \(i\) is the imaginary unit (i.e., \(i^2 = -1\)).
- **Denominator \((2n)!\)**: Factorial of \(2n\), which is the product of all positive integers up to \(2n\).
This series resembles the Taylor expansion of trigonometric or hyperbolic functions involving complex numbers.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2c222075-437b-48bc-845a-e38a8bd40737%2Fe75fe44a-c390-427e-9a43-ca707f2811a2%2F6uo8b93_processed.png&w=3840&q=75)
Transcribed Image Text:The image presents the mathematical expression for an infinite series:
\[
\sum_{n=0}^{\infty} \frac{(-1)^n (1+i)^{2n}}{(2n)!}
\]
### Components of the Expression:
- **Sigma Notation (\(\sum\))**: Represents the summation of terms from \(n = 0\) to infinity (\(\infty\)).
- **\((-1)^n\)**: Alternating sign factor, which changes the sign of each term depending on whether \(n\) is even or odd.
- **\((1+i)^{2n}\)**: The base \(1+i\) is raised to the power of \(2n\). Here, \(i\) is the imaginary unit (i.e., \(i^2 = -1\)).
- **Denominator \((2n)!\)**: Factorial of \(2n\), which is the product of all positive integers up to \(2n\).
This series resembles the Taylor expansion of trigonometric or hyperbolic functions involving complex numbers.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
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,
