Let
∑
𝑛
=
1
∞
(
−
1
)
𝑛
𝑎
𝑛
be a series where

∀
𝑛
,

𝑎
𝑛

0

∀
𝑛
,

𝑎
𝑛

𝑎
𝑛
+
1

lim
𝑛
→
∞
𝑎
𝑛
=
0

Define
𝑆
=
lim
𝑘
→
∞
𝑆
𝑘
where
𝑆
𝑘
=
∑
𝑛
=
1
𝑘
(
−
1
)
𝑛
𝑎
𝑛
.
Given
𝑆
5
=
3.5
,

𝑆
6
=
3.6
,

𝑆
7
=
3.56
,

𝑆
8
=
3.58
.

Which of the following statements MUST be true?

Select all the correct answers.

Question

Let 
∑
𝑛
=
1
∞
(
−
1
)
𝑛
𝑎
𝑛
  be a series where

∀
𝑛
,
 
𝑎
𝑛
>
0
  
∀
𝑛
,
 
𝑎
𝑛
>
𝑎
𝑛
+
1
  
lim
𝑛
→
∞
𝑎
𝑛
=
0
 
Define  
𝑆
=
lim
𝑘
→
∞
𝑆
𝑘
   where   
𝑆
𝑘
=
∑
𝑛
=
1
𝑘
(
−
1
)
𝑛
𝑎
𝑛
 .
Given 
𝑆
5
=
3.5
,
 
𝑆
6
=
3.6
,
 
𝑆
7
=
3.56
,
 
𝑆
8
=
3.58
 .

Which of the following statements MUST be true?

Select all the correct answers.

BlackTom AI · Accepted Answer

Let us restate the problem and the given statements to ensure clarity before evaluating each option.

Question context: We have a series ∑_{n=1}^∞ (-1)^n a_n with a_n > 0 for all n, a_n > a_{n+1} for all n (i.e., the sequence a_n is positive and decreasing), and lim_{n→∞} a_n = 0. Define S_k = ∑_{n=1}^k (-1)^n a_n and S = lim_{k→∞} S_k (the alternating sum limit, if it exists).

Given values: S_5 = 3.5, S_6 = 3.6, S_7 = 3.56, S_8 = 3.58. We are asked which statements MUST be true.

Option A: |S_5 − S| < 0.1
- This asks whether the distance from the partial sum at n=5 to the limit S is guaranteed to be less than 0.1.
- We know only S_5 = 3.5 and the later partial sums approach S. While S_6, S_7, S_8 suggest the sequence of partial sums is approaching a value around 3.57–3.58, the given data do not strictly bound the eventual limit S within 0.1 of S_5. It could be, or could not be, within 0.1; the information is insufficient to guarantee this bound. Therefore, this statement cannot be established as MUST be true from the given data.

Option B: ∑_{n=1}^∞ (-1)^n a_n is convergent.
- Here we have a_n > 0 for all n, a_n > a_{n+1} for all n, and lim a_n = 0. The sequence (a_n) is positive and monotonically decreasing to 0. The alternating series test (Leibniz criterion) says that if the terms decrease to zero, the alternating series ∑ (-1)^n a_n converges. Since all the hypotheses for the test are satisfied, this series is convergent. Thus, this statement MUST be true.

Option C: a_7 = −0.04
- Since it is given that a_n > 0 for all n, every term a_n is positive. An equality stating a_7 equals a negative number (−0.04) directly contradicts a_n > 0. Therefore, this statement is false and cannot be true.

Option D: |S_7 − S| > 0.02
- This asks whether the distance between S_7 and the limit S must exceed 0.02.
- From the data: S_7 = 3.56, and the surrounding partial sums suggest S is near 3.57–3.58. The actual distance |S_7 − S| could be around 0.01–0.02, depending on the exact limit S which we do not know exactly from the given values. Because the limit could be as close as about 0.01–0.02 away, we cannot guarantee that the distance is greater than 0.02. Thus, this statement does not MUST be true.

In summary:
- Option A cannot be confirmed as true from the provided information.
- Option B follows directly from the monotone decreasing to zero nature of a_n and the alternating sign, so it MUST be true.
- Option C is impossible given a_n > 0, so it is false.
- Option D cannot be guaranteed true with the limited partial-sum data, so it is not MUST be true.

Therefore, the statement that MUST be true among the options is the convergence of the alternating series in Option B.

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