题目
题目

MTH1030 -1035 - S1 2025 MTH1030/35 Week 10 lesson quiz: Power series defining functions

单项选择题

Puzzle for you. Let \(s\) be a convergent series. The series consisting of the positive terms of \(s\) is \(s^+\) and the series consisting of the negative terms is \(s^-\). Which of the following statements is not always true.

选项
A.a. If both \(s^+\) and \(s^-\) diverge to infinity, then no matter what your favorite number is, \(s\) can be rearranged into a series with your favourite number as its sum.
B.b. If \(s^+\) diverges to infinity and \(s^-\) has a finite sum, then \(s\) is absolutely convergent.
C.c. If both \(s^+\) and \(s^-\) diverge to infinity, then \(s\) is conditionally convergent.
D.d. If both \(s^+\) and \(s^-\) diverge to infinity, then \(s\) can be rearranged into a series that diverges.
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思路分析
We start by restating the problem and listing the options for clarity. Question: Puzzle for you. Let s be a convergent series. The series consisting of the positive terms of s is s^+ and the series consisting of the negative terms is s^-. Which of the following statements is not always true? Answer options: - a. If both s^+ and s^- diverge to infinity, then no matter what your favorite number is, s can be rearranged into a series with your favourite number as its sum. - b. If s^+ diverges to infinity and s^- has a finite sum, then s is absolutely convergent. - c. If both s^+ and s^- diverge to infinity, then s is conditionally convergent. - d. If both s^+ and s^- diverge to infinity, then s can be rearranged into a series that diverges. Now, let’s analyze each option carefully, considering what each statement asserts about the positive and negative parts of a convergent series and the consequences for rearrangements and convergence. Option a: - Claim: If both s^+ and s^- diverge to infinity, then you can rearrange the terms of s so that the sum equals any chosen real number (your favorite number). - What this hinges on: This resembles the Riemann rearrangement phenomenon......Login to view full explanation

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