Problem 41 - College Algebra

Question

Find the sum of each infinite geometric series. $$1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots$$

Step-by-Step Solution

Answer

The sum of the infinity geometric series is \( \frac{2}{3} \)

Step 1: Identify the first term (a) and the ratio (r)

In this problem, the first term a is 1 and the ratio r is -1/2. You can see the ratio by dividing any term in the sequence by the preceding term.

Step 2: Check if the series is convergent

The series is convergent if the absolute value of r is less than 1. In this case, absolute value of r is |-1/2| = 1/2, which is less than 1. Therefore, the series is convergent and has a sum.

Step 3: Apply the formula for the sum of an infinite geometric series

The formula is \( S = \frac{a}{1 - r} \). Substituting the values we have, \( S = \frac{1}{1 - (-1/2)} = \frac{1}{1 + 1/2} = \frac{1}{3/2} = \frac{2}{3} \).

Key Concepts

Convergence of Series

When we talk about series in mathematics, specifically an infinite series, one of the first questions we ask is whether the series converges or diverges. A series converges if the sum of its terms approaches a certain number as more terms are added. For an infinite geometric series, such as \(1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \cdots\), the condition for convergence is that the absolute value of the common ratio (r) must be less than 1.

Why does this matter? Well, if the absolute value of r is 1 or greater, the terms do not get progressively smaller, and so, they cannot add up to a finite number. In our example, the ratio is \(-\frac{1}{2}\) and its absolute value is \(\frac{1}{2}\), which is less than 1. This indicates the series converges and has a sum that we can calculate.

Geometric Series Formula

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The formula for the sum of a finite geometric series is already well-known, but when we deal with an infinite geometric series and we know it converges, we use a slightly different formula to find the sum: \( S = \frac{a}{1 - r} \), where S is the sum of the series, a is the first term, and r is the common ratio.

This elegant formula simplifies the process of summing an infinite number of terms, as long as the necessary condition of |r| < 1 is met. By examining the given series and identifying the first term and the common ratio, we can apply this formula directly to find the sum. It's a powerful tool in mathematics, often revealing surprising and beautiful relationships in what might first appear to be a complex series.

Summation of Series

The summation of a series involves adding all terms of the series together. In an infinite geometric series with a convergent condition, this will result in a definitive sum. The process for finding the sum can be understood by looking at how the series terms get smaller when the absolute value of r is less than one, and how they eventually approach zero.

The sum can be thought of as the 'limit' of the series as the number of terms goes to infinity. In our example, the procedure to find the sum of the series was systematic: First, we identified the first term and the ratio, then we verified the convergence, and finally, we used the convergence fact and the geometric series formula to find that the sum of the series \(1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \cdots\) is \(\frac{2}{3}\). Understanding this process is crucial to grasping the overall concept of summation for infinite geometric series.