Understanding Series Convergence and Divergence

This video tutorial explains how to determine if a series converges or diverges. It begins by distinguishing between sequences and series, then introduces the concept of partial sums and infinite series. The tutorial provides methods to determine convergence or divergence, including taking limits and using the divergence test. Examples are given to illustrate these concepts, focusing on arithmetic series and the application of the divergence test.

What is the main difference between a sequence and a series?

• A sequence is a list of numbers, while a series is the sum of a sequence.

• A sequence and a series are the same.

• A sequence is always finite, while a series is always infinite.

• A sequence is a sum of numbers, while a series is a list of numbers.

What does the term 'partial sum' refer to in the context of series?

• The sum of all terms in a sequence.

• The product of terms in a series.

• The difference between two terms in a sequence.

• The sum of a finite number of terms in a series.

How can you determine if an infinite series converges?

• By calculating the sum of the first ten terms.

• By finding a finite limit for the partial sums as n approaches infinity.

• By checking if the sequence of terms is increasing.

• By ensuring the sequence of terms is decreasing.

What does it mean if the limit of the partial sums of a series is infinity?

• The series converges to a finite number.

• The series diverges.

• The series is undefined.

• The series converges to zero.

What is the divergence test used for?

• To calculate the limit of a sequence.

• To quickly identify if a series diverges.

• To find the sum of a series.

• To determine if a sequence converges.

According to the divergence test, what happens if the limit of a sequence as n approaches infinity is not zero?

• The series converges to zero.

• The series is undefined.

• The series diverges.

• The series converges.

In the example problem, what is the sequence a_n given as?

• n^2

• 7n - 4

• 2n

• 5n + 3

What is the result of applying the divergence test to the sequence 2n?

• The series converges.

• The series converges to zero.

• The series diverges.

• The series is undefined.

What is the limit of the sequence 5n + 3 over 7n - 4 as n approaches infinity?

• 0

• 1

• 5/7

• Infinity

Why does the series 5n + 3 over 7n - 4 diverge?

• Because the sequence is decreasing.

• Because the sequence is increasing.

• Because the limit of the sequence is zero.

• Because the limit of the sequence is not zero.