Determine whether the series converges absolutely or conditionally, or diverges. (-1)^ 20 00 n = 1
Determine whether the series converges absolutely or conditionally, or diverges. (-1)^ 20 00 n = 1
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
![**Problem Statement:**
Determine whether the series converges absolutely or conditionally, or diverges.
\[
\sum_{n=1}^{\infty} \frac{(-1)^n}{2^n}
\]
**Explanation:**
This is an alternating series where the general term is given by:
\[
a_n = \frac{(-1)^n}{2^n}
\]
To determine the convergence, we can use the Alternating Series Test. For a series \(\sum_{n=1}^{\infty} (-1)^n b_n\):
1. \(b_n\) must be positive.
2. \(b_n\) must be decreasing.
3. \(\lim_{n \to \infty} b_n = 0\).
In this case, \(b_n = \frac{1}{2^n}\), which is positive, decreasing, and approaches zero as \(n\) approaches infinity.
Additionally, to determine absolute convergence, we check the convergence of the series of absolute values:
\[
\sum_{n=1}^{\infty} \left|\frac{(-1)^n}{2^n}\right| = \sum_{n=1}^{\infty} \frac{1}{2^n}
\]
This is a geometric series with a common ratio \(r = \frac{1}{2}\), which is less than 1. Geometric series converge if \(|r| < 1\). Therefore, the series converges absolutely.
**Conclusion:**
The series converges absolutely.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F45235a53-69d6-46fa-83a8-de0bf8bf66d1%2Fe8944549-f7ce-4abb-93a7-9d0ff178791a%2Fnum3gmf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Determine whether the series converges absolutely or conditionally, or diverges.
\[
\sum_{n=1}^{\infty} \frac{(-1)^n}{2^n}
\]
**Explanation:**
This is an alternating series where the general term is given by:
\[
a_n = \frac{(-1)^n}{2^n}
\]
To determine the convergence, we can use the Alternating Series Test. For a series \(\sum_{n=1}^{\infty} (-1)^n b_n\):
1. \(b_n\) must be positive.
2. \(b_n\) must be decreasing.
3. \(\lim_{n \to \infty} b_n = 0\).
In this case, \(b_n = \frac{1}{2^n}\), which is positive, decreasing, and approaches zero as \(n\) approaches infinity.
Additionally, to determine absolute convergence, we check the convergence of the series of absolute values:
\[
\sum_{n=1}^{\infty} \left|\frac{(-1)^n}{2^n}\right| = \sum_{n=1}^{\infty} \frac{1}{2^n}
\]
This is a geometric series with a common ratio \(r = \frac{1}{2}\), which is less than 1. Geometric series converge if \(|r| < 1\). Therefore, the series converges absolutely.
**Conclusion:**
The series converges absolutely.
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 2 steps with 2 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,
