题目
题目
单项选择题

Puzzle for you. Let [math: s] be a convergent series. The series consisting of the positive terms of [math: s] is [math: s+]s^+ and the series consisting of the negative terms is [math: s−]s^-. Which of the following statements is not always true.

选项
A.a. If both [math: s+]s^+ and [math: s−]s^- diverge to infinity, then no matter what your favorite number is, [math: s] can be rearranged into a series with your favourite number as its sum.
B.b. If [math: s+]s^+ diverges to infinity and [math: s−]s^- has a finite sum, then [math: s] is absolutely convergent.
C.c. If both [math: s+]s^+ and [math: s−]s^- diverge to infinity, then [math: s] is conditionally convergent.
D.d. If both [math: s+]s^+ and [math: s−]s^- diverge to infinity, then [math: s] can be rearranged into a series that diverges.
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思路分析
The question presents a convergent series s and defines s+ as the series of its positive terms and s− as the series of its negative terms. It then asks which of the statements is not always true, so we must examine each option for its universal validity. Option 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 that sum. This aligns with the Riemann rearrangement ideas: when both the positive and negative parts have infinite sums, the original series is not absolutely convergent and can be rearranged to yield a wide range o......Login to view full explanation

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