Understanding Alternating Series Tests

The video tutorial explains the concept of alternating series and the alternating series test. It provides examples of alternating series, such as the alternating harmonic series, and demonstrates how to apply the alternating series test to determine convergence or divergence. The video also covers the divergence test and provides multiple examples to illustrate the concepts. Advanced examples are discussed, and the video concludes with a summary of the key points.

What is the primary characteristic of an alternating series?

• It has terms that are all positive.

• It alternates between positive and negative terms.

• It has terms that are all zero.

• It has terms that are all negative.

Which condition must be met for a series to pass the divergence test?

• The limit of a_n as n approaches infinity must equal zero.

• The limit of a_n as n approaches infinity must be less than zero.

• The limit of a_n as n approaches infinity must be greater than zero.

• The limit of a_n as n approaches infinity must be infinite.

In the alternating harmonic series, what is the sequence a_n?

• n^2

• (-1)^n

• 1/n

• (-1)^n * n

What happens to the series if the limit of a_n as n approaches infinity is not zero?

• The series oscillates.

• The series becomes constant.

• The series diverges.

• The series converges.

How does the factorial series 1/n! behave as n approaches infinity?

• It diverges to infinity.

• It converges to zero.

• It oscillates.

• It remains constant.

What is the result of applying the alternating series test to the series 1/5^n?

• The series diverges.

• The series oscillates.

• The series is undefined.

• The series converges.

When using L'Hopital's Rule, what is the derivative of ln(n)?

• ln(n)

• n

• 1/n

• 1

In the series ln(n)/n, what happens as n approaches infinity?

• The series becomes constant.

• The series converges to zero.

• The series oscillates.

• The series diverges to infinity.

What is the role of cosine in the series involving cosine(nπ)/n?

• To create the alternating effect.

• To make the series negative.

• To make the series positive.

• To make the series constant.

What is the conclusion about the series cosine(nπ)/n?

• The series oscillates.

• The series converges.

• The series diverges.

• The series is undefined.