Question

Explain how the Root Test works.

Short answer

Question: Explain the Root Test and its possible outcomes for determining the convergence or divergence of an infinite series. Answer: The Root Test is used to determine the convergence or divergence of an infinite series by analyzing the limit of the nth root of the absolute value of the terms as n approaches infinity. If the limit (L) is less than 1, the series converges, meaning the terms approach zero quickly enough for the series to converge to a finite value. If L is greater than 1, the series diverges, meaning the terms do not decay quickly enough to result in a finite sum. If L equals 1 or the limit does not exist, the test is inconclusive, and another test (such as the Ratio Test or Comparison Test) should be used to determine whether the series converges or diverges.

Step-by-step solution

Definition of the Root Test

The Root Test can be used to determine the convergence or divergence of an infinite series. The test states that for an infinite series ∑a_n, consider the limit L:= lim(n->∞) (|a_n|)^(1/n). If 0≤L<1, the series converges, if L>1, the series diverges, and if L=1 or the limit does not exist, the test is inconclusive.

Scenario for convergence

If the limit of the nth root of the absolute value of the terms in the series, L, is less than 1, then the series converges. This means that the terms approach zero quickly enough so that their sum (the series) converges to a finite value.

Scenario for divergence

If the limit L is greater than 1, then the series diverges. This means that the terms of the series do not decay quickly enough to make the series converge to a finite value. Instead, the terms will contribute to an infinite sum.

Scenario for inconclusive outcomes

If the limit L is equal to 1 or does not exist, the Root Test provides no information about the convergence or divergence of the series. In this case, we must use another test (such as the Ratio Test or the Comparison Test) to determine if the series converges or diverges.

Applying the Root Test

To apply the Root Test, you need to find the limit L:= lim(n->∞) (|a_n|)^(1/n). Calculate the absolute value of the terms in the series, take the nth root of the result, and find the limit of this expression as n approaches infinity. Then, compare the limit to 1 and use the results from Steps 2, 3, and 4 to determine the convergence or divergence of the series.