Understanding Alternating Series and Divergence

The video tutorial explains the alternating series test, which determines if an infinite series converges or diverges. It reviews the conditions for convergence, such as the limit of terms approaching zero and non-increasing terms. An example series is analyzed, showing that it diverges because the limit of its terms does not equal zero. Detailed calculations are provided to illustrate the process, and the video concludes by verifying that the series fails the necessary conditions for convergence.

What is the primary condition for an alternating series to converge?

• The terms must be increasing.

• The series must be finite.

• The limit of the terms must be zero.

• The terms must be positive.

In the given example, what is the non-alternating part of the series?

• n to the ninth plus two divided by n to the fifth

• n to the fifth divided by n to the ninth

• n to the fifth divided by the square root of n to the ninth plus two

• n to the ninth divided by n to the fifth

How do we determine the limit of a series by comparing degrees?

• By comparing the highest degree terms in numerator and denominator

• By adding the degrees of numerator and denominator

• By multiplying the degrees of numerator and denominator

• By subtracting the degree of the denominator from the numerator

What happens to the series if the limit of its terms is not zero?

• The series converges

• The series diverges

• The series becomes finite

• The series oscillates

What is the result of dividing the numerator by n to the nine halves in the example?

• n to the power of three

• n to the power of zero

• n to the power of two

• n to the power of one half

Why does the alternating series test not apply in this example?

• The limit of the terms is not zero

• The terms are increasing

• The series is finite

• The terms are not positive

What is the approximate value of a sub one in the example?

• 0.577

• 1.992

• 3.000

• 2.500

What does it indicate if a sub n plus one is greater than a sub n?

• The series is constant

• The series is finite

• The series diverges

• The series converges

What is the approximate value of a sub three in the example?

• 1.992

• 0.577

• 3.000

• 2.500

What is the final conclusion about the series in the example?

• The series converges

• The series diverges

• The series is finite

• The series is constant