Problem 52 - Algebra and Trigonometry

Question

Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. $$ 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots $$

Step-by-Step Solution

Answer

The series is convergent with a sum of \(\frac{2}{3}\).

Step 1: Identify the first term and common ratio

The series given is: \(1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \cdots\). The first term \(a\) of the series is \(1\). The series alternates in sign, indicating a common ratio \(r\). To find the common ratio, divide the second term by the first term: \(r = -\frac{1}{2} \div 1 = -\frac{1}{2}\).

Step 2: Determine convergence of the series

A geometric series converges if the absolute value of the common ratio is less than 1, i.e., \(|r| < 1\). Here, \(|r| = \left|-\frac{1}{2}\right| = \frac{1}{2}\), which is less than 1. Therefore, the series is convergent.

Step 3: Calculate the sum of the series

The sum \(S\) of an infinite geometric series can be found using the formula \(S = \frac{a}{1 - r}\), where \(a\) is the first term, and \(r\) is the common ratio. Substituting the known values: \(S = \frac{1}{1 - (-\frac{1}{2})} = \frac{1}{1 + \frac{1}{2}} = \frac{1}{\frac{3}{2}} = \frac{2}{3}\).

Key Concepts

Convergence

In mathematics, convergence is a crucial concept, especially when dealing with series and sequences. Simply put, a series is considered convergent when its terms slow down and approach a specific number as the series progresses. If this doesn't happen, the series is said to diverge.
For a geometric series, convergence depends primarily on the common ratio, denoted by \( r \). A geometric series will converge if the absolute value of \( r \), written as \(|r|\), is less than 1. This means the terms get smaller and smaller, eventually leading to a specific sum, rather than wandering off to infinity.
In the exercise provided, the series is convergent because the absolute value of the common ratio, \(|r| = \frac{1}{2}\), is clearly less than 1.

Infinite Series

An infinite series is a sum of infinitely many terms. To imagine this, think of stacking blocks one on top of the other without ever stopping. Some infinite series may seem daunting because of the never-ending addition. However, many of them "settle down" at a particular value, which is the essence of convergence.
Not all infinite series have a sum. For example, the series of natural numbers \(1 + 2 + 3 + \dots\) never converge to a single value. They just keep growing. But a geometric series, as long as the common ratio \(|r|\) is less than 1, has a unique sum.

• Convergent: sums to a specific value.
• Divergent: doesn’t sum up to a single number.

For the given series, since it is convergent, it indeed has a finite sum, calculated to be \(\frac{2}{3}\).

Common Ratio

In any geometric series, the common ratio is the factor that you multiply each term by to get to the next term. This is the constant between the terms of the sequence. Think of it as the recipe step to add ingredients (terms) consistently.
The common ratio \( r \) can be either positive or negative:

• If \( r \) is positive, all terms have the same sign.
• If \( r \) is negative, the terms will alternate in sign as seen in the exercise with the first few terms: \(1, -\frac{1}{2}, \frac{1}{4}, \dots\).

To find the common ratio, divide any term by its preceding term. For the series \(1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \cdots\), the common ratio \( r \) is calculated as \(-\frac{1}{2} \div 1 = -\frac{1}{2}\). Understanding this concept helps in determining both the nature and behavior of the series. It sets the pace and direction (convergent or divergent) of the whole calculation.