Convergence Tests for Series

Professor Dave explains alternating series, where terms alternate in sign. He introduces the alternating series test to determine convergence, using examples to illustrate convergent and divergent series. The video also covers absolute and conditional convergence, and the ratio test for absolute convergence.

What is a key characteristic of an alternating series?

• The terms are all equal.

• All terms are positive.

• The signs of the terms alternate between positive and negative.

• All terms are negative.

What is the purpose of the alternating series test?

• To check if a series is geometric.

• To determine if a series is arithmetic.

• To assess whether an alternating series converges or diverges.

• To find the sum of a series.

Which of the following is a criterion for the convergence of an alternating series?

• The terms must increase in absolute value.

• The terms must decrease in absolute value and approach zero.

• The terms must all be negative.

• The terms must all be positive.

What does it mean if a series is absolutely convergent?

• The series diverges when the absolute value of each term is taken.

• The series converges only when all terms are positive.

• The series converges when the absolute value of each term is taken.

• The series converges only when all terms are negative.

What is the difference between absolute and conditional convergence?

• Absolute convergence means the series converges when absolute values are taken, while conditional convergence means it converges without absolute values.

• Absolute convergence means the series converges with positive terms, while conditional convergence means it converges with negative terms.

• There is no difference between absolute and conditional convergence.

• Absolute convergence means the series diverges, while conditional convergence means it converges.

What is the ratio test used for?

• To determine if a series is arithmetic.

• To find the sum of a series.

• To determine if a series is absolutely convergent.

• To check if a series is geometric.

In the ratio test, what does it mean if the limit L is less than one?

• The series is absolutely convergent.

• The series is conditionally convergent.

• The series is arithmetic.

• The series is divergent.

What happens if the limit L equals one in the ratio test?

• The series is absolutely convergent.

• The ratio test is inconclusive.

• The series is conditionally convergent.

• The series is divergent.

What is Euler's number approximately equal to?

• 1.61

• 3.14

• 1.41

• 2.72

In the example using the ratio test, what was the conclusion about the series N to the N power over N factorial?

• The series is absolutely convergent.

• The series is arithmetic.

• The series is conditionally convergent.

• The series is divergent.