Problem 66 - Precalculus: Mathematics for Calculus

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 infinite geometric series is given as \(1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \cdots\). The first term \(a\) is \(1\), and the common ratio \(r\) can be found by dividing the second term by the first term: \(r = -\frac{1}{2}\).

Step 2: Determine convergence or divergence

For a geometric series to converge, the common ratio \(|r|\) must be less than 1. In this series, \(|r| = \left| -\frac{1}{2} \right| = \frac{1}{2}\), which is less than 1. Therefore, the series is convergent.

Step 3: Use the formula for the sum of a convergent series

The sum \(S\) of an infinite geometric series can be calculated with the formula \(S = \frac{a}{1 - r}\), where \(a\) is the first term and \(r\) is the common ratio. Substitute \(a = 1\) and \(r = -\frac{1}{2}\) into the formula: \[ S = \frac{1}{1 - (-\frac{1}{2})} = \frac{1}{1 + \frac{1}{2}} = \frac{1}{\frac{3}{2}} = \frac{2}{3}.\]

Key Concepts

Convergent Series

A convergent series is a series whose terms get closer and closer to a specific value as the number of terms increases. In the context of geometric series, convergence occurs when the series settles into a stable sum. For a geometric series to be convergent, the absolute value of the common ratio must be less than one.

• This means that the terms decrease in size and the series approaches a finite limit.
• If \(|r| < 1\), the series is convergent. Otherwise, it is divergent.

In our example, the series \(1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \cdots\) is convergent because the common ratio \(|r| = \frac{1}{2}\), which is less than one.
Understanding convergence is important because it helps us determine whether the series has a sum that can be calculated.

Infinite Series

An infinite series is essentially the sum of an infinite number of terms. This can initially be an abstract concept because adding up infinitely many numbers hints at an undefined or infinite result. However, some infinite series can indeed have a finite sum, and these are particularly interesting because they challenge our preconception of infinity.
When dealing with a geometric series like \(1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \cdots\), it’s the behavior dictated by the common ratio that determines the convergence.

• If the common ratio signaling a reduction in terms is observed, the series can be summed to a specific value despite being infinite.
• If the series is divergent with \(|r| \geq 1\), it doesn’t settle on a particular sum.

Infinite series are a crucial concept in calculus and analysis, used to represent a broad range of mathematical and real-world phenomena.

Common Ratio

The common ratio in a geometric series is a vital element that helps determine the nature of the series. It is found by dividing any term in the series by the previous term. This ratio is consistent throughout the series, dictating how successive terms relate.
In the given series, the common ratio \(r\) is \(-\frac{1}{2}\).

• The formula to find subsequent terms is multiplying the last term by the common ratio.
• Because the \(|r| \) is less than one, we can conclude that this series is convergent.
• The behavior of this ratio, being negative, also causes the series to alternate in sign.

Understanding the common ratio lets us quickly identify whether the geometric series will converge, and assists in calculating its sum using mathematical formulas.