Question

Which of the following statements is true?

a. The Root Test says that the series converges absolutely.
b. The Root Test says that the series diverges.
c. The Root Test says that the series converges conditionally.
d. The Root Test is inconclusive, but the series converges absolutely by another test or tests.
e. The Root Test is inconclusive, but the series diverges by another test or tests.
f. The Root Test is inconclusive, but the series converges conditionally by another test or tests.

Answer 1

The Root Test is used to determine the convergence or divergence of an infinite series based on the limit of the nth root of the absolute terms. Option d is correct: The Root Test is inconclusive, but the series converges absolutely by another test or tests.

Step-by-step explanation:

The Root Test is a convergence test used to determine the convergence or divergence of an infinite series. It involves taking the nth root of the absolute value of the terms in the series and then evaluating the limit as n approaches infinity. Based on the limit value, different conclusions can be made.

If the limit is less than 1, then the series converges absolutely.

If the limit is greater than 1 or infinity, then the series diverges.

If the limit is equal to 1 or is inconclusive, the Root Test is inconclusive, and further tests must be used to determine the convergence or divergence of the series.

Therefore, option d is the correct statement: The Root Test is inconclusive, but the series converges absolutely by another test or tests.

Answer 2

Answer: L = 3/4

Option A. The Root Test says that the series converges absolutely.

Step-by-step explanation:

By using the root test equation given in the question. L = 3/4

Since L < 1, the series converges absolutely.

For clarity of expression, the detailed calculation is contained in the attached file. Check the file attached for the complete calculation to this question.

Step-by-step explanation: