Convergence Tests for Series

The video tutorial explains how to determine if an infinite series converges absolutely, converges conditionally, or diverges. It begins by defining absolute and conditional convergence. The first series is analyzed using the limit comparison test, showing it diverges. The second series is examined with the alternating series test, revealing it is conditionally convergent. The tutorial provides a step-by-step approach to applying these tests, emphasizing the importance of understanding the behavior of series terms.

What is the condition for a series to be absolutely convergent?

• The series itself converges.

• The series diverges.

• The series and its absolute value both converge.

• The series is conditionally convergent.

Which test is applied to determine the divergence of the first series?

• Ratio Test

• Limit Comparison Test

• Integral Test

• Alternating Series Test

In the limit comparison test, what is the significance of the limit being greater than zero?

• The series is conditionally convergent.

• The series diverges.

• The series converges.

• The series is absolutely convergent.

What is the main characteristic of an alternating series?

• All terms are positive.

• Terms are constant.

• Terms alternate in sign.

• All terms are negative.

What does the alternating series test check for?

• Divergence

• Convergence of an alternating series

• Conditional convergence

• Absolute convergence

What must be true for a series to be conditionally convergent?

• The series diverges.

• The series is absolutely convergent.

• The series and its absolute value both converge.

• The series converges, but its absolute value diverges.

How is the limit comparison test used to determine divergence in the second series?

• By comparing with a divergent series

• By using the integral test

• By using the ratio test

• By comparing with a convergent series

What is the result of the limit comparison test for the absolute value of the second series?

• The series is conditionally convergent.

• The series is absolutely convergent.

• The series diverges.

• The series converges.

What does it mean if the absolute value of a series diverges but the series itself converges?

• The series is neither convergent nor divergent.

• The series is conditionally convergent.

• The series is absolutely convergent.

• The series diverges.

What is the conclusion about the original infinite series in the last section?

• It is conditionally convergent.

• It diverges.

• It is absolutely convergent.

• It is neither convergent nor divergent.