Give the definition of the sum of a series of complex numbers, and show that the Comparison Test continues to hold: if {z} is sequence of complex numbers and{b is a sequence of nonnegative numbers such that z ≤ b for all n, and if Σ b converges, then Σ z converges n=1 n=1
Give the definition of the sum of a series of complex numbers, and show that the Comparison Test continues to hold: if {z} is sequence of complex numbers and{b is a sequence of nonnegative numbers such that z ≤ b for all n, and if Σ b converges, then Σ z converges n=1 n=1
Chapter9: Sequences, Probability And Counting Theory
Section9.4: Series And Their Notations
Problem 4SE: How is finding the sum of an infinite geometric series different from finding the nth partial sum?
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![**Definition and Comparison Test for Complex Series**
**Objective:**
To define the sum of a series of complex numbers and demonstrate the application of the Comparison Test to complex series.
**Introduction:**
Consider a series \( \sum_{n=1}^{\infty} z_n \), where \( \{z_n\} \) is a sequence of complex numbers. To establish convergence of such series, we utilize the Comparison Test.
**Comparison Test Statement:**
Suppose \( \{z_n\} \) is a sequence of complex numbers and \( \{b_n\} \) is a sequence of nonnegative real numbers, satisfying the following condition:
\[ |z_n| \leq b_n \text{ for all } n \]
If the series \( \sum_{n=1}^{\infty} b_n \) converges, then the series \( \sum_{n=1}^{\infty} z_n \) also converges.
**Explanation:**
- \( |z_n| \) represents the magnitude of the complex number \( z_n \).
- The condition \( |z_n| \leq b_n \) implies that the series of magnitudes \( \sum |z_n| \) is bounded above by the convergent series \( \sum b_n \).
- Thus, if \( \sum b_n \) converges, by the Comparison Test, the series \( \sum z_n \) converges as well.
This principle reinforces how convergence properties of real-number sequences can extend to complex-number sequences, underlining the versatility of the Comparison Test in complex analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F37e8ed93-7bef-4409-89ed-52264f64a27e%2Ff9c4ce2f-5285-4146-8c20-369b75608f0d%2Frt7feal_processed.png&w=3840&q=75)
Transcribed Image Text:**Definition and Comparison Test for Complex Series**
**Objective:**
To define the sum of a series of complex numbers and demonstrate the application of the Comparison Test to complex series.
**Introduction:**
Consider a series \( \sum_{n=1}^{\infty} z_n \), where \( \{z_n\} \) is a sequence of complex numbers. To establish convergence of such series, we utilize the Comparison Test.
**Comparison Test Statement:**
Suppose \( \{z_n\} \) is a sequence of complex numbers and \( \{b_n\} \) is a sequence of nonnegative real numbers, satisfying the following condition:
\[ |z_n| \leq b_n \text{ for all } n \]
If the series \( \sum_{n=1}^{\infty} b_n \) converges, then the series \( \sum_{n=1}^{\infty} z_n \) also converges.
**Explanation:**
- \( |z_n| \) represents the magnitude of the complex number \( z_n \).
- The condition \( |z_n| \leq b_n \) implies that the series of magnitudes \( \sum |z_n| \) is bounded above by the convergent series \( \sum b_n \).
- Thus, if \( \sum b_n \) converges, by the Comparison Test, the series \( \sum z_n \) converges as well.
This principle reinforces how convergence properties of real-number sequences can extend to complex-number sequences, underlining the versatility of the Comparison Test in complex analysis.
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