Question

Each of the following statements is an attempt to show that a given series is convergent or divergent using the comparison test. for each statement, enter c (for "correct") if the argument is valid, or enter i (for "incorrect") if any part of the argument is flawed. (note: if the conclusion is true but the argument that led to it was wrong, you must enter i.) i 1. for all , , and the series converges, so by the comparison test, the series converges. i 2. for all , , and the series diverges, so by the comparison test, the series diverges. i 3. for all , , and the series converges, so by the comparison test, the series converges. c 4. for all , , and the series converges, so by the comparison test, the series converges. i 5. for all , , and the series diverges, so by the comparison test, the series diverges. c 6. for all , , and the series converges, so by the comparison test, the series converges. note: to get full credit, all answers must be correct. having all but one correct is worth 50%. two or more incorrect answers gives a score of 0%.

Answer 1

The correctness of arguments based on the comparison test relies on the proper application of the test's rules against a known benchmark series, to determine whether a given series converges or diverges.

Step-by-step explanation:

The use of the comparison test in determining the convergence or divergence of a series relies on comparing it to a known benchmark series. Each statement provided by the student proposes an argument based on the comparison test and concludes whether the given series converges or diverges. These arguments should be assessed for their validity. For example, if a series with non-negative terms is compared to a convergent benchmark series with larger terms, then the original series also converges. Conversely, if a series with non-negative terms is compared to a divergent series with smaller terms, then the original series also diverges.

However, the validity of these arguments depends not only on the truth of the comparison but also on the appropriate application of the comparison test and the series' behavior.

The correctness of each argument must be analyzed in the context of the comparison test's rules, even if the conclusion (convergence or divergence) is correct.