Understanding the Divergence Test

The video tutorial explains the divergence test, a method to determine if a series diverges. It covers the test's main idea: if the limit of a sequence as n approaches infinity is not zero, the series diverges. Examples include a series with n/(3n-4), the harmonic series, and a geometric series. The video also discusses when a series may converge or diverge, emphasizing the need for additional tests if the limit equals zero. The tutorial concludes with a geometric series example, highlighting conditions for convergence.

What is the main purpose of the divergence test?

• To determine if a series converges

• To calculate the limit of a sequence

• To find the sum of a series

• To determine if a series diverges

In the example with the sequence n/(3n-4), what is the limit as n approaches infinity?

• 2/3

• 1

• Infinity

• 0

Why does the harmonic series diverge?

• Because it is a geometric series

• Because the terms increase

• Because the terms decrease but the sum does not converge

• Because the limit of the sequence is not zero

What is the common ratio of the geometric series 1/(2^n)?

• 1

• 0

• 1/2

• 2

For a geometric series to converge, what must be true about the common ratio?

• It must be less than 1

• It must be equal to 1

• It must be negative

• It must be greater than 1

What is the sum of the geometric series 1/(2^n) as n approaches infinity?

• 0

• 1/2

• 1

• Infinity

In the series 8/5^n, what happens to the limit as n approaches infinity?

• It approaches infinity

• It remains constant

• It approaches zero

• It approaches one

What is the common ratio of the series 8/5^n?

• 1

• 0

• 8/5

• 5/8

Why does the series 8/5^n diverge?

• Because the limit is zero

• Because the terms decrease

• Because the common ratio is greater than 1

• Because the common ratio is less than 1

What can be concluded if the limit of a sequence is not zero?

• The series diverges

• The series converges

• The series is geometric

• The series has a finite sum