Use the sequence feature of a graphing calculator to graph...

Use the sequence feature of a graphing calculator to graph the first ten terms of each sequence as defined. Use the graph to make a conjecture as to whether the sequence converges or diverges. If it converges, determine the number to which it converges. $$a_{n}=\left(1+\frac{1}{n}\right)^{n}$$

Key Concepts

Sequence

A sequence is an ordered list of numbers defined by a specific rule. In many mathematical contexts, sequences can be analyzed term-by-term, and properties such as convergence or divergence are determined by examining the behavior of its terms as the index increases.

Convergence of a Sequence

The concept of convergence in sequences refers to whether the terms of a sequence approach a specific value as the index increases indefinitely. A sequence converges if its terms eventually get arbitrarily close to a particular number, called the limit, which represents the value that the sequence is approaching.

Limit

The limit of a sequence is the value that the terms of the sequence approach as the index becomes very large. Deciding on the limit involves determining whether the sequence settles at one number. If a limit exists, the sequence is said to converge; if not, it diverges.

Euler's Number (e)

Euler's number, denoted as 'e', is a fundamental mathematical constant approximately equal to 2.71828. It arises naturally in contexts involving growth processes and compound interest, and it is defined as the limit of the sequence (1 + 1/n)^n as n approaches infinity.

Graphing Calculators

Graphing calculators provide a visual way to analyze sequences by plotting their individual terms. By graphing the terms of a sequence, one can visually assess patterns such as convergence, divergence, or periodic behavior. This graphical approach aids in forming conjectures about the behavior of the sequence and supports analytical methods.

Transcript

00:01 All right, in this video, we'll be looking at a sequence using a graphic calculator.

00:05 I'm using desmos right here to kind of show you what's going to happen with the graph.

00:08 And we're also looking at the first 10 terms and then come up with the conclusion if the sequence converges or diverges.

00:13 Okay, just to kind of refresh your memory, converge means that the graph or function actually goes to a number, approaches the same number.

00:20 If it diverges, then the graph may be going in different directions, opposite directions, could be going to positive directions.

00:28 Could be going to a positive negative infinity.

00:29 It's not going to a particular number.

00:31 Okay.

00:32 So that's what we're going to look at and kind of look at it graphically.

00:34 Okay, i'm going to show what the graph looks like right now.

00:40 Okay, now one thing we want to do when we look at the graph is we're only going to look at where it goes to positive infinity or basically the right side of the graph because that's how the term numbers work in a sequence is when it's going to positive infinity.

00:49 So even though there's a graph on the left hand side, we can't really look at that.

00:52 So if you look on the right hand side, if you notice it's kind of leveling out, it's obviously approaching something.

00:57 I just don't know what that is yet.

00:58 Okay.

00:59 So we're going to hand and look at our table of values.

01:01 To kind of figure out what that answer is.

01:03 All right.

01:03 So right here, i'm just plugging in one, two, three, up to ten, just kind of see what numbers i have, and then i'll kind of show you from there what we can do next.

01:10 So i got one, two, three, four, five, six, seven, eight, nine, ten.

01:21 All right.

01:22 So if you look at the numbers right here in the table, obviously they're getting, they're different, but they're getting really close to something...

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