Problem 24 - Essential Calculus

Question

Verify that the infinite series converges. $$ \sum_{n=0}^{\infty}(-0.6)^{n}=1-0.6+0.36-0.216+\cdots $$

Step-by-Step Solution

Answer

The infinite series converges and the sum is 0.625

Step 1: Identify terms

Recognize the series as a geometric series with \(a=1\) and \(r=-0.6\). The sum of an infinite geometric series \(\sum_{n=0}^{\infty}ar^n\) is only defined if the common ratio \(r\) has an absolute value less than 1.

Step 2: Check the condition

Check whether the absolute value of \(r\) is less than 1. In this case, \(|r|=|-0.6|=0.6\) which is less than 1. Therefore, the series converges.

Step 3: Compute the sum

Use the formula for the sum of an infinite geometric series, which is \(S=a/(1-r)\), where \(S\) is the sum of the series. Substituting \(a=1\) and \(r=-0.6\), the sum of this series is \(S=1/(1-(-0.6))=1/1.6=0.625\).

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This type of series can be easily recognized by this repeating pattern of multiplication. For instance, in the series \(1, -0.6, 0.36, -0.216, \ldots\) each term is \(0.6\) times the term before it.

Geometric series are significant in mathematics because they are one of the simplest types of infinite series with rules and formulas for determining their behavior, such as convergence and summation.

Sum of an Infinite Geometric Series

The sum of an infinite geometric series can be calculated with the formula \( S = \frac{a}{1 - r} \), where \(S\) is the sum, \(a\) is the first term, and \(r\) is the common ratio. However, this formula only applies when the series is convergent, which means the terms are getting smaller and approaching zero. For example, in the series \(1 - 0.6 + 0.36 - 0.216 + \cdots\), using the formula, the sum is \( S = \frac{1}{1 - (-0.6)} = \frac{1}{1.6} = 0.625 \).

The power of this formula lies in its ability to provide a finite value for the summation of an infinitely long series, allowing us to understand and work with infinite processes.

Common Ratio

The common ratio in a geometric series is the constant factor between consecutive terms. In the provided series, every term is obtained by multiplying the preceding term by \( -0.6 \) (the common ratio). The common ratio plays a crucial role in determining the properties of the series, such as whether it converges or diverges. For instance, if the common ratio's absolute value is greater than one, the series diverges, as the magnitude of the terms will increase indefinitely. Conversely, if the absolute value is less than one, the series terms decrease in magnitude and add up to a finite sum.

Series Convergence Criteria

In order for a geometric series to converge, the absolute value of the common ratio must be less than one (\(|r| < 1\)). This criterion is derived from the need for the series terms to get progressively smaller. In our example, we verify that the absolute value of the common ratio \(\left|-0.6\right|\) is indeed less than one, which confirms that the series converges.

Moreover, the convergence criteria are crucial to determining the behavior of the series because they ensure that as more terms are added, their contribution to the total sum decreases, eventually leading to a definable limit. This criterion is integral to the proper application of the sum formula for an infinite geometric series.