Understanding Sequence Convergence

The video tutorial explores the convergence or divergence of the sequence a_n = (-1)^n/n. It explains the concept of limits and how they determine convergence. The absolute value theorem for sequences is introduced, showing that if the limit of the absolute value of a sequence is zero, the sequence itself converges to zero. The tutorial includes a step-by-step calculation and graphical representation of the sequence and its absolute value, concluding that the sequence converges.

What is the sequence a_n defined as in the video?

• a_n = (-1)^n / n

• a_n = n^(-1)

• a_n = n / (-1)^n

• a_n = n^2

What is the condition for a sequence to be considered convergent?

• The sequence must be positive.

• The sequence must oscillate.

• The limit must be a constant.

• The limit must be infinite.

What theorem is used to determine the convergence of the sequence?

• Pythagorean Theorem

• Absolute Value Theorem

• Binomial Theorem

• Fundamental Theorem of Calculus

According to the absolute value theorem, if the limit of the absolute value of a sequence is zero, what can be said about the original sequence?

• It becomes infinite.

• It diverges.

• It converges to zero.

• It oscillates.

What is the absolute value of (-1)^n?

• 1

• -1

• n

• 0

What is the limit of 1/n as n approaches infinity?

• Negative infinity

• One

• Zero

• Infinity

How does the graph of the sequence a_n differ from its absolute value?

• The graph of a_n is a straight line.

• The graph of a_n is always above the x-axis.

• The graph of the absolute value is below the x-axis.

• The graph of the absolute value flips negative values above the x-axis.

What happens to the negative values in the graph of the absolute value of the sequence?

• They become zero.

• They become positive infinity.

• They remain negative.

• They flip above the x-axis.

What conclusion is drawn about the sequence a_n using the absolute value theorem?

• The sequence diverges.

• The sequence oscillates indefinitely.

• The sequence converges to zero.

• The sequence becomes infinite.

Why does the absolute value theorem make sense for sequences?

• Because it only applies to positive sequences.

• Because it only applies to sequences with negative terms.

• Because if the absolute value converges, the original sequence must also converge.

• Because it is based on the Pythagorean theorem.