Problem 18 Graph the sequence. $$\left\\{... [FREE SOLUTION]

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Precalculus: Functions and Graphs

Earl W. Swokowski, Jeffrey A. Cole

$Math Studyset 91影视 Explanations$ Math

12 Edition

Chapter 9: Problem 18

Graph the sequence. $$\left\\{\frac{1}{n}\right\\}$$

Short Answer

Expert verified

Plot points (1, 1), (2, 0.5), (3, 0.333), etc., and draw a curve approaching the x-axis.

Step by step solution

Understand the Sequence

The sequence given is $ \left\{ \frac{1}{n} \right\} $ where $ n $ starts from 1 and goes to infinity. This means that each term of the sequence is the reciprocal of the natural numbers.

Create a Table of Values

To plot the sequence, it's often helpful to create a table of values for the first few terms. For example, when $ n = 1 $, the term is $ \frac{1}{1} = 1 $. When $ n = 2 $, the term is $ \frac{1}{2} = 0.5 $. Continue this for several terms: - $ n = 3 $, $ \frac{1}{3} \approx 0.333 $ - $ n = 4 $, $ \frac{1}{4} = 0.25 $ - $ n = 5 $, $ \frac{1}{5} = 0.2 $ and so on.

Plot Points on Coordinate Plane

On a coordinate plane, the $ x $-axis represents the term number $ n $, and the $ y $-axis represents the value of each term $ \frac{1}{n} $. Plot the points you have from the table of values: - (1, 1) - (2, 0.5) - (3, 0.333) - (4, 0.25) - (5, 0.2) and continue for the values you calculated.

Draw the Graph

Once the points are plotted, draw a smooth curve that connects the points. The curve should approach the $ x $-axis (which is the line $ y = 0 $) as $ n $ increases, showing that the sequence terms get closer to zero, but never actually reach it.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reciprocal Sequences

A reciprocal sequence involves taking the reciprocal of each term of a sequence. For the sequence $ \left\{ \frac{1}{n} \right\} $, each term is the reciprocal of a natural number. This means:

For $ n = 1 $, $ \frac{1}{1} = 1 $.
For $ n = 2 $, $ \frac{1}{2} = 0.5 $.
For $ n = 3 $, $ \frac{1}{3} \approx 0.333 $.
And so on, continuing for larger values of $ n $.

The concept of a reciprocal is simple: each number flips over. For example, the reciprocal of 5 is $ \frac{1}{5} = 0.2 $. As $ n $ gets larger, $ \frac{1}{n} $ becomes smaller. This illustrates how reciprocal sequences start large and become smaller, showing an interesting mathematical behavior as they progress.

Coordinate Plane Plotting

Plotting a sequence on a coordinate plane provides a visual representation of its behavior. To graph the sequence $ \left\{ \frac{1}{n} \right\} $, labelling of axes is crucial:

The $ x $-axis represents the term number $ n $.
The $ y $-axis represents the value of each term $ \frac{1}{n} $.

Here's how to plot the points:

Start with the first term: plot the point $(1, 1)$.
Proceed with the second term: mark $(2, 0.5)$.
Continue plotting $(3, 0.333), (4, 0.25), (5, 0.2)$, and so on.

Joining these points smoothly, you begin to see how the line curves downwards, approaching the $ x $-axis. This visualization helps to better understand how the values progress and how they approach zero but never actually touch the $ x $-axis.

Convergence to Zero

In the sequence $ \left\{ \frac{1}{n} \right\} $, as the term number $ n $ becomes larger, the value of each term $ \frac{1}{n} $ becomes smaller. This phenomenon is known as **convergence to zero**. Here's how it happens:

When $ n $ is small, the value, such as $ 1 $ or $ 0.5 $, is relatively large.
As $ n $ grows, the value decreases: $ 0.333 $, $ 0.25 $, $ 0.2 $, and continues.
The terms become insignificantly small as $ n $ grows larger and larger.

The sequence gets incredibly close to zero, but it will never actually be zero, as long as $ n $ remains a positive number. Mathematically, this is expressed as: $ \lim_{{n \to \infty}} \frac{1}{n} = 0 $. This central idea in calculus shows how sequences can approach a limit without necessarily reaching it in finite steps.

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91影视

Graph the sequence. $$\left\\{\frac{1}{n}\right\\}$$

Short Answer

Step by step solution

Understand the Sequence

Create a Table of Values

Plot Points on Coordinate Plane

Draw the Graph

Key Concepts

Reciprocal Sequences

Coordinate Plane Plotting

Convergence to Zero

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