Converging & Diverging Sequences - Calculus 2

We are now making our way into the next major group of concepts in Calculus 2, which is sequences and series. We will start by first introducing sequences, which is the least complicated of the two. 

In mathematics, a sequence is an ordered list of terms. A very basic example would be a list of all the positive integers beginning with 1: {1, 2, 3, 4, 5, ...}. This sequence would continue on forever, with each term increasing by 1.

Sequences can be represented by an expression called the "nth term of the sequence." Essentially, it is an expression in terms of a variable n that represents each term of the sequence for a specific value of n, where n is assumed to only be positive integers starting with 1. You can view n for a sequence as the equivalent of x in a typical function f(x). It represents the input values that will produce the output values, which are the terms of the sequence.

We are interested in sequences with regards to calculus because we can use calculus to determine if the terms of a sequence are heading in a particular direction. In other words, we can figure out if the terms of a sequence are converging to a specific value as n gets increasingly larger. For example, consider this sequence: {1, 0.5, 0.25, 0.125, 0.0625, ...} where each subsequent term gets closer to 0. We can see that the terms of this sequence are slowly becoming 0, but we can show that this is true by using a limit. By taking the limit of the nth term of a sequence, we can determine the "convergence" of a sequence. Thus, once you have been introduced to what sequences are, this concept of a converging sequence is the main focus of this new lesson in Calculus 2. Now let's get to it!

In this lesson, you will learn:

• What a sequence is in mathematics, and the different types of sequences
• How to write out the terms of a sequence and express a list of terms as a sequence
• How to determine the convergence of a sequence by using limits

I hope you find this video to be helpful!

-Josh