Problem 29 - College Algebra

Question

Use the geometric series \(\sum_{k=1}^{\infty}\left(\frac{1}{2}\right)^{k}\). What number does \(S_{n}\) seem to be approaching in the graph? Find the sum to explain why this makes sense.

Step-by-Step Solution

Answer

The series converges to 1.

Step 1: Understand the Problem

We are given an infinite geometric series \(\sum_{k=1}^{\infty} \left(\frac{1}{2}\right)^{k}\) and need to find the sum it converges to as more terms are added. We need to analyze what sum this series approaches.

Step 2: Identify the Formula

The sum \(S\) of an infinite geometric series with first term \(a\) and common ratio \(r\), where \(-1 < r < 1\), is given by the formula \(S = \frac{a}{1-r}\).

Step 3: Determine 'a' and 'r'

In our series, the first term \(a = \left(\frac{1}{2}\right)^{1} = \frac{1}{2}\) and the common ratio \(r = \frac{1}{2}\).

Step 4: Apply the Formula

Using the formula for the sum of an infinite geometric series: \[ S = \frac{\frac{1}{2}}{1 - \frac{1}{2}} = \frac{\frac{1}{2}}{\frac{1}{2}} = 1 \]Thus, the series converges to 1.

Step 5: Interpret the Graph

If the graph plots the partial sums \(S_n = \sum_{k=1}^{n} \left(\frac{1}{2}\right)^{k}\) against \(n\), it will show a curve approaching the line \(y = 1\) as \(n\) increases, verifying our calculation.

Key Concepts

Infinite Series

In mathematics, an infinite series is the sum of the terms of an infinite sequence. One well-known type of infinite series is a geometric series, which is a series where each term is a constant multiple of the previous term. The general form is \[ S = a + ar + ar^2 + ar^3 + \cdots \]where:

• \(a\) is the first term
• \(r\) is the common ratio

This series goes on forever, adding term after term. However, even though it is made up of infinitely many terms, it can often sum up to a finite number, which is a fascinating idea.
A specific example is the series \(\sum_{k=1}^{\infty} \left(\frac{1}{2}\right)^{k}\). In this series, each term is half of the previous one. It starts from \(\frac{1}{2}\) and continues without end.

Convergence of Series

Convergence is a key concept when dealing with infinite series. A series is said to converge to a limit if the sequence of its partial sums approaches a specific value as more terms are added. For an infinite geometric series to converge, the absolute value of the common ratio \(r\) must be less than 1. This is expressed as \[ -1 < r < 1 \]When this condition is met, the series sum can be calculated with the formula: \[ S = \frac{a}{1-r} \]where \(a\) is the first term.
In the given series \(\sum_{k=1}^{\infty} \left(\frac{1}{2}\right)^{k}\), we have \(a = \frac{1}{2}\) and \(r = \frac{1}{2}\). Since \(|\frac{1}{2}| < 1\), the series is convergent. Using the formula, we find that it converges to 1, meaning the sum of all its infinite terms is equal to 1

Partial Sums

Partial sums are used to understand how an infinite series behaves as you add more terms. A partial sum, denoted as \(S_n\), is the sum of the first \(n\) terms of a series. In a geometric series like \(\sum_{k=1}^{\infty} \left(\frac{1}{2}\right)^{k}\), if you calculate \(S_n\) for increasingly larger values of \(n\), you will begin to see that \(S_n\) gets closer and closer to the limit, or the convergent sum.
In our series, the partial sums would look like:

• \(S_1 = \frac{1}{2}\)
• \(S_2 = \frac{3}{4} = \frac{1}{2} + \frac{1}{4}\)
• \(S_3 = \frac{7}{8} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8}\)

Each \(S_n\) is an approximation of the infinite sum, and as \(n\) increases, \(S_n\) becomes a better approximation of the sum 1. If plotted on a graph, the curve of these partial sums would approach the line \(y = 1\), illustrating how the series converges.