Let
∑
𝑛
=
1
∞
𝑎
𝑛
be a series. Note:
𝑎
𝑛
is not necessary to be positive.

Which of the following statements MUST be true?

Select all the correct answers.

Question

Let 
∑
𝑛
=
1
∞
𝑎
𝑛
 be a series. Note: 
𝑎
𝑛
 is not necessary to be positive.

Which of the following statements MUST be true?

Select all the correct answers.

BlackTom AI · Accepted Answer

We start by restating the setup in plain terms: we have a series sum from n=1 to ∞ of a_n, where a_n may be negative or positive. We are asked to evaluate which of the given statements must be true for all such sequences {a_n}.

Option 1: If ∑ |a_n| converges, then ∑ a_n converges.
- This is a standard result: absolute convergence implies convergence. When the series of absolute values is convergent, the original series is absolutely convergent, and absolute convergence guarantees convergence of the series itself (by the comparison/majorant principle and Cauchy criteria). Therefore, this statement should hold for any sequence {a_n} with ∑ |a_n| convergent.

Option 2: If ∑ a_n is divergent, then lim_{n→∞} a_n ≠ 0.
- A fundamental property of series is that if ∑ a_n converges, then a_n → 0. The contrapositive is: if a_n does not tend to 0, the series diverges. However, the contrapositive does not directly address the case when the series diverges: it is possible for a divergent series to have a_n → 0 (e.g., harmonic series ∑ 1/n has a_n → 0 but diverges). So this statement is false because there exist divergent series with a_n → 0. Thus, this must-not-be-true in general.

Option 3: If ∑ a_n is convergent, then ∑ |a_n| is convergent.
- This is the reverse of absolute convergence. A convergent series need not be absolutely convergent. There are many conditional convergent series (for example, the alternating harmonic series ∑ (-1)^{n+1}/n converges, but ∑ |(-1)^{n+1}/n| = ∑ 1/n diverges). Hence, this statement is false in general.

Option 4: If ∑ a_n is divergent, then ∑ a_n = ∞.
- This statement asserts that divergence implies the sum diverges to positive infinity. But divergent series can diverge to negative infinity, oscillate without limit, or behave in other non-convergent ways. For instance, ∑ (-1)^n does not converge and does not tend to ∞; it oscillates. Therefore, this statement is incorrect because divergences are not universally equal to +∞.

Option 5: If ∑ a_n is divergent, then ∑ |a_n| is divergent.
- Consider a divergent series where a_n alternates in sign but with decreasing magnitude, like a_n = (-1)^n/n. The original series ∑ a_n converges (it is the alternating harmonic series), but if we take a divergent example, we should test whether divergence of ∑ a_n forces divergence of ∑ |a_n|. However, there exist divergent ∑ a_n where ∑ |a_n| diverges as well (many faster-decreasing sequences with sign changes still yield divergence in the absolute sum). But the statement as written is claiming that whenever ∑ a_n diverges, the sum of absolute values must also diverge. This is generally true because if ∑ |a_n| converged (i.e., is finite), then by Option 1, ∑ a_n would converge, which would contradict divergence. Therefore, whenever ∑ a_n diverges, ∑ |a_n| cannot converge; hence ∑ |a_n| must diverge. This reasoning shows Option 5 is necessarily true.

Summarizing the assessments of each option:
- Option 1: True (absolute convergence implies convergence of the original series).
- Option 2: False (a_n can tend to 0 while ∑ a_n diverges, so lim a_n ≠ 0 is not required).
- Option 3: False (convergence of ∑ a_n does not guarantee convergence of ∑ |a_n|; conditional convergence is possible).
- Option 4: False (divergence is not necessarily to +∞; it can be −∞ or oscillatory).
- Option 5: True (if ∑ a_n diverges, ∑ |a_n| cannot converge, otherwise Option 1 would force convergence of ∑ a_n).

In essence, the must-be-true statements are those that align with the core principles of absolute convergence and the contrapositive relationship between convergence of ∑ |a_n| and convergence of ∑ a_n, while rejecting claims that rely on incorrect implications about the nature of divergence or about the necessity of absolute convergence.

Let ∑ 𝑛 = 1 ∞ 𝑎 𝑛 be a series. Note: 𝑎 𝑛 is not necessary to be positive. Which of the following statements MUST be true? Select all the correct answers.多项选择题

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