Understanding the Divergence Test

The video tutorial explains the divergence test, which determines if an infinite series diverges based on the limit of its terms. It highlights that the test cannot confirm convergence. Through three examples, the video demonstrates how to apply the test by analyzing the degrees of the numerator and denominator, calculating limits, and using graphical verification. The first example shows a series that might converge or diverge, the second confirms divergence due to a higher degree numerator, and the third also confirms divergence with equal degrees but different leading coefficients.

What does the divergence test help determine about an infinite series?

• Whether the series converges

• Whether the series diverges

• The exact sum of the series

• The initial terms of the series

If the limit of a series' terms as n approaches infinity is zero, what can we conclude?

• The series is finite

• The series might converge or diverge

• The series diverges

• The series converges

In the first example, what is the degree of the numerator in the series?

• Four

• One

• Two

• Three

What happens to the terms of the series in the first example as n approaches infinity?

• They remain constant

• They become larger

• They approach zero

• They oscillate

In the second example, why does the series diverge?

• The numerator's degree is lower

• The series is finite

• The numerator's degree is higher

• The denominator's degree is higher

What is the limit of the series in the second example as n approaches infinity?

• Two-thirds

• Zero

• One

• Infinity

In the third example, what is the ratio of the leading coefficients?

• Three-fourths

• Four-fifths

• Two-thirds

• One-half

What conclusion can be drawn from the third example's limit?

• The series diverges

• The series is finite

• The series is undefined

• The series converges

How does the graph of the terms in the third example behave?

• It approaches zero

• It oscillates

• It approaches infinity

• It stabilizes at two-thirds

What is the significance of the limit being non-zero in the third example?

• It indicates a finite series

• It indicates oscillation

• It indicates divergence

• It indicates convergence