Problem 84 - College Algebra

Question

Many graphing utilities have a sequence-graphing mode that plots the terms of a sequence as points on a rectangular coordinate system. Consult your manual; if your graphing utility has this capability, use it to graph each of the sequences . What appears to be happening to the terms of each sequence as \(n\) gets larger? $$a_{n}=\frac{n}{n+1} \quad n:[0,10,1] \text { by } a_{n}:[0,1,0.1]$$

Step-by-Step Solution

Answer

As \(n\) gets larger, the value of the sequence \(a_{n}\) seems to approach 1, as evidenced by the graph. This indicates that, in the limit as \(n\) goes to infinity, the sequence \(a_{n}=\frac{n}{n+1}\) tends to 1.

Step 1: Computation of sequence values

To start with, compute the values of \(a_{n}\) for the provided range of \(n\). This is done by substituting each \(n\) value from 0 through 10 (in increments of 1) into the formula \(a_{n}=\frac{n}{n+1}\).

Step 2: Generation of sequence graph

Upon obtaining the sequence values, plot the values of \(n\) along the horizontal axis and the corresponding \(a_{n}\) values along the vertical axis to create the graph of the sequence.

Step 3: Analysis of Sequence Behaviour

Examine the graph and observe what happens to the terms of the sequence \(a_{n}\) as \(n\) gets larger. The relationships and patterns noticed inform conclusions about the behaviour of the sequence as \(n\) grows larger.

Key Concepts

Sequence Convergence

When studying sequences, a fundamental concept is sequence convergence. Convergence means that as you progress through the terms of a sequence, they start getting closer and closer to a specific value. This particular value is commonly referred to as the limit of a sequence.

Imagine you're walking towards a wall; with each step, you halve the distance to the wall. You're getting infinitely closer to the wall, yet you never really touch it. This is akin to a sequence that gets infinitely close to a number, which would be its limit. In the case of the sequence presented in the exercise, \(a_{n}=\frac{n}{n+1}\), as \(n\) grows larger, the terms appear to approach 1; they never quite reach it but get closer and closer. This suggests that the sequence converges to 1.

Rectangular Coordinate System

A rectangular coordinate system is a 2-D grid formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). Points on this grid are defined by pairs of numbers, which represent how far along each axis the point lies.

For sequences, we often put the term number, or the \(n\)-value, on the x-axis, and the value of the term itself, or \(a_{n}\), on the y-axis. Graphically, this helps students visualize the behavior of a sequence's terms as \(n\) increases. In the exercise, plotting our sequence on such a system allows us to observe convergence visually by looking at the pattern and direction of the plotted points.

Limits of a Sequence

The limit of a sequence is the value that the terms of a sequence get closer to as \(n\) increases without bound. Mathematically, we say that the sequence \(a_{n}\) has a limit \(L\) if the terms of \(a_{n}\) can get arbitrarily close to \(L\) for all sufficiently large values of \(n\).

For example, in the given sequence \(a_{n}=\frac{n}{n+1}\), you can observe that as \(n\) becomes larger, the difference between \(a_{n}\) and 1 becomes smaller and smaller. If we say that as \(n\) approaches infinity, \(a_{n}\) approaches 1, we express it as \(\lim_{n \to \infty} a_{n} = 1\). This notion of limiting behavior is key to understanding many other concepts in calculus and higher mathematics.

Graphing Utilities

In our digital age, graphing utilities—software or calculators that can plot graphs—are invaluable tools for visualizing sequences and their behaviors. These utilities take a sequence's formula and range for \(n\), compute the values, and plot them on a rectangular coordinate system.

Using a sequence-graphing mode, students can see patterns and draw conclusions that might not be immediately evident from the numerical data alone. For instance, by graphing the sequence \(a_{n}=\frac{n}{n+1}\) within the range of \(n=[0,10,1]\), students quickly notice that the plotted points draw nearer to the line \(y=1\), hinting at its limit. This graphical representation helps in understanding convergence and other properties by providing a visual context.