Root Test for Series Convergence

The video tutorial explains the Root Test for determining the convergence or divergence of an infinite series. It describes the conditions under which a series converges, diverges, or when the test fails. The tutorial includes three examples demonstrating the application of the Root Test on different types of series, highlighting the importance of the nth root and exponent manipulation in the process.

What does the Root Test determine about an infinite series when the limit L is less than 1?

• The series oscillates.

• The series converges.

• The series diverges.

• The test is inconclusive.

If the limit L equals 1 in the Root Test, what should be done?

• The series diverges.

• The series converges.

• A different test should be applied.

• The series is undefined.

In the first example, what is the result of applying the Root Test to the series?

• The test is inconclusive.

• The series is undefined.

• The series converges.

• The series diverges.

What is the significance of the leading coefficients in the first example?

• They show the series is undefined.

• They indicate divergence.

• They are irrelevant to the test.

• They determine the limit at infinity.

In the second example, what happens to the limit as n approaches infinity?

• It approaches zero.

• It approaches positive infinity.

• It remains constant.

• It becomes negative.

What is the outcome of the Root Test in the second example?

• The series is undefined.

• The test is inconclusive.

• The series diverges.

• The series converges.

In the third example, what is the fixed value of the numerator as n approaches infinity?

• e to the fifth

• e to the fourth

• e cubed

• e squared

What is the result of the Root Test in the third example?

• The series diverges.

• The series converges.

• The test is inconclusive.

• The series is undefined.

What is the general conclusion about the Root Test from the lesson?

• It is always conclusive.

• It is useful for series with terms raised to the nth power.

• It is only applicable to geometric series.

• It is not a reliable test.

What should be done if the Root Test is inconclusive?

• Apply a different convergence test.

• Assume the series converges.

• Assume the series diverges.

• Apply the Ratio Test.