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.

MAT187H1 F LEC0101 Integral Test for Series (Pre-Class Essentials)

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

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