Using the last question, show that if {n}_₁ and {wn}_₁ are complex numbers such ∞ that 1ns and ₁wn = t for some complex numbers s and t, then Σ(²n+wn) = 8 + t ∞ n=1 ∞ n=1
Using the last question, show that if {n}_₁ and {wn}_₁ are complex numbers such ∞ that 1ns and ₁wn = t for some complex numbers s and t, then Σ(²n+wn) = 8 + t ∞ n=1 ∞ 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
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Question
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Please solve part b only
![### Problem Statement
**a)** Suppose \(\{z_n\}_{n=1}^\infty\) and \(\{w_n\}_{n=1}^\infty\) are sequences of complex numbers such that \(\lim_{n \to \infty} z_n = z\) and \(\lim_{n \to \infty} w_n = w\) for some complex numbers \(z\) and \(w\). Show that
\[
\lim_{n \to \infty} (z_n + w_n) = z + w
\]
**b)** Using the last question, show that if \(\{z_n\}_{n=1}^\infty\) and \(\{w_n\}_{n=1}^\infty\) are complex numbers such that \(\sum_{n=1}^\infty z_n = s\) and \(\sum_{n=1}^\infty w_n = t\) for some complex numbers \(s\) and \(t\), then
\[
\sum_{n=1}^\infty (z_n + w_n) = s + t
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa62f7b35-7db6-46d8-92c3-a45ad2747ea7%2F4411d266-3352-45ae-8380-5f66ab817684%2Fzgzm7ii_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**a)** Suppose \(\{z_n\}_{n=1}^\infty\) and \(\{w_n\}_{n=1}^\infty\) are sequences of complex numbers such that \(\lim_{n \to \infty} z_n = z\) and \(\lim_{n \to \infty} w_n = w\) for some complex numbers \(z\) and \(w\). Show that
\[
\lim_{n \to \infty} (z_n + w_n) = z + w
\]
**b)** Using the last question, show that if \(\{z_n\}_{n=1}^\infty\) and \(\{w_n\}_{n=1}^\infty\) are complex numbers such that \(\sum_{n=1}^\infty z_n = s\) and \(\sum_{n=1}^\infty w_n = t\) for some complex numbers \(s\) and \(t\), then
\[
\sum_{n=1}^\infty (z_n + w_n) = s + t
\]
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