Which of the following series can you use the integral test to determine if they converge? Select all that apply.

A:
B:
C:
D:

Question

Which of the following series can you use the integral test to determine if they converge? Select all that apply.

A: 
B: 
C: 
D:

BlackTom AI · Accepted Answer

These questions hinge on whether the integral test is applicable to a given series, so the essential task is to inspect the form of the terms and the behavior of the corresponding function.
First, note the key criterion for the integral test: you have a series of the form sum from n=1 to infinity of a_n, where a_n = f(n) for some continuous, positive, decreasing function f defined on [1, ∞). If such an f exists and the improper integral from 1 to infinity of f(x) dx converges, then the series converges; if the integral diverges, so does the series.
Next, even without the actual explicit terms, there are common situations where the integral test can be applied:
- When a_n = 1/n^p with p > 0 and f(x) = 1/x^p is positive, continuous, and decreasing for x ≥ 1; the integral test yields convergence iff p > 1.
- When a_n = 1/log(n) or a_n = 1/(n log n), the integral test can be used but requires careful comparison to the corresponding integral; often these diverge or converge based on known benchmarks like the integral of 1/x and 1/(x log x).
- When a_n is not monotone or becomes negative, the standard integral test does not apply directly, and alternative convergence tests are more appropriate.
Now, addressing the statement that options C and D are correct: since the actual series forms for options C and D are not provided in the prompt, we cannot map those letters to specific series or verify their eligibility for the integral test. Without the explicit terms or the function forms, we cannot determine whether each option meets the necessary conditions (a_n = f(n) with f continuous, positive, decreasing on [1, ∞)).
If you supply the explicit series terms for options A through D, I can walk through each one and apply the integral test step by step:
- Check whether a_n can be written as f(n) with f(x) positive, continuous, decreasing for x ≥ some point.
- If so, consider the improper integral ∫ from 1 to ∞ f(x) dx and determine convergence or divergence.
- If the series fails the monotonicity or positivity requirement, explain why the integral test is not appropriate and suggest alternative tests (e.g., comparison test, limit comparison test, ratio test, or alternating series test) as needed.
In summary, the integral test applies to series of the form sum f(n) with f positive, continuous, decreasing on [N, ∞). The lack of explicit option content prevents a concrete verdict on C and D, but the above criteria provide a clear checklist to use once the actual series terms are available.

Which of the following series can you use the integral test to determine if they converge? Select all that apply. A: B: C: D: 多项选择题

类似问题

You are trying to determine whether diverges. Which integral can you use to give a lower bound to the sequence of partial sums ? A: B: C: D:

Let 𝑓 be a CONTINUOUS, POSITIVE, DECREASING function on [ 0 , ∞ ) . Which of the following statements must be true? Select all the correct answers.

What is the result of applying the integral test to the series [math: ∑n=1∞12n−1] \displaystyle\sum\limits_{n=1}^ \infty \frac{1}{2n-1} ?

The gradient of the straight line, which passes through (3, 5) and (5, 3) is:

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单