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.
$$a_{n}=2 e^{n}$$

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Key Concepts

Graphical Analysis of Sequences

Graphing sequences using tools such as graphing calculators helps in visualizing the behavior of a sequence. By plotting the terms, one can visually assess whether the sequence converges to a limit or diverges, making it a practical method for conjecturing about its long-term behavior.

Exponential Growth

Exponential functions are characterized by their rapid rate of increase as the exponent increases. In the context of sequences, those defined by exponential expressions typically exhibit exponential growth, which can lead to divergence if the growth is unbounded.

Convergence and Divergence

The concepts of convergence and divergence describe the long-term behavior of a sequence. A convergent sequence approaches a specific finite number as the index increases, while a divergent sequence does not settle to a finite limit, often growing without bound or oscillating.

Sequence Notation

A sequence is an ordered list of numbers where each term is generated by a specific rule or formula. Studying sequences involves understanding how the terms evolve as the index increases and is foundational in calculus and discrete mathematics.

Transcript

00:01 All right, in this video, we're looking at a sequence that we're given, but kind of figuring out the answer to a graph and calculator.

00:06 So if using desmond's right here to kind of show you, it's asked us to find with our calculator, graph and calculator to find the first 10 terms.

00:13 And then kind of using that information to conclude if the sequence it converges or diverges.

00:18 So just to kind of refresh your converge means that a sequence as it goes to infinity, because that's how sequences go.

00:23 It's from, on the positive end of the term numbers.

00:29 If it converges, it's going to a number.

00:32 It's kind of getting really close to something graphically.

00:34 If it diverges, that could mean that the sequence kind of splits graphically or it's going to positive or negative infinity.

00:39 It's not really going to a number.

00:40 So in order for something to converge, it has to be going or approaching a number.

00:44 Okay, diverges meaning it doesn't approach a single number or a particular number.

00:48 All right.

00:48 So right here i have the function already kind of plugged in right here.

00:51 I am going to tap this to show what it looks graphically.

00:54 Now, if i look at the graph, we're going to positive infinity, notice that it's shooting up...

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