Problem 72 - Precalculus: Mathematics for Calculus

Question

Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. $$1-1+1-1+\cdots$$

Step-by-Step Solution

Answer

The series is divergent because the common ratio is -1, causing oscillation.

Step 1: Identify the First Term and Common Ratio

The given series is \(1 - 1 + 1 - 1 + \cdots\). The first term, \(a\), of this series is 1. To find the common ratio \(r\), divide the second term by the first term: \(-1 / 1 = -1\). Thus, the common ratio \(r = -1\).

Step 2: Determine Convergence Criteria for an Infinite Geometric Series

An infinite geometric series converges if the absolute value of the common ratio \(|r|\) is less than 1. This ensures that the terms get smaller and smaller, approaching zero, allowing a finite sum to be calculated.

Step 3: Check the Convergence Condition

For our series, the common ratio \(r = -1\). Calculate \(|r|\): \(|-1| = 1\). Since \(|r| = 1\) is not less than 1, the series does not satisfy the condition for convergence.

Step 4: Conclusion on Convergence

Since the series has a common ratio of -1, the sequence of terms does not decay to zero and oscillates indefinitely as \(1, -1, 1, -1, \cdots\). Therefore, the series is divergent and does not have a sum.

Key Concepts

Convergence and Divergence

Infinite geometric series can either be convergent or divergent, and understanding these terms is essential to determine the behavior of the series over an infinite number of terms. Convergence refers to the tendency of a series' terms to approach zero, resulting in a finite sum. A convergent series eventually gets closer and closer to a certain value as more terms are added.
In contrast, a divergent series does not settle towards any finite number. Its terms might oscillate or increase indefinitely. The series given in the exercise, for instance, diverges because its terms keep alternating between 1 and -1. There is no consistent approach to zero or any finite sum.

• Convergence: Terms approach zero. The series can sum to a finite number.
• Divergence: Terms do not approach zero or a finite value cannot be determined.

Understanding whether a series converges or diverges helps determine if a finite sum is possible.

Common Ratio

The common ratio in a geometric series measures how each term relates to the one before it. It’s defined by dividing any term in the series by the previous term. For the series in the problem, the common ratio was calculated as follows: divide -1 by 1 to obtain -1, which is the common ratio.
The common ratio is crucial because it helps determine convergence. Specifically, a series converges when the absolute value of the common ratio, denoted as \(|r|\), is less than one. This means that the terms are getting progressively smaller. However, if \(|r|\) is equal to or greater than one, like in this series where \(|r|\) is one, it indicates the series is divergent.

• Common Ratio (\

Sum of a Series

While finding the sum of a series is a straightforward task for convergent series, it becomes impossible for divergent series. Divergent series, like the alternating series of 1 and -1 in this problem, do not settle into a single sum. As such, there’s no straightforward formula for their sum. For convergent series with common ratio \(|r| < 1\), the sum can be calculated using a specific formula:
\[ S = \frac{a}{1-r} \]
where \( S \) represents the sum, \( a \) is the first term, and \( r \) is the common ratio. This formula won’t apply to the example due to its divergence, emphasizing that only convergent series allow for a finite summation.

• Sum Formula: Only valid for convergent series
• Divergent Series: No finite sum can be determined

Recognizing the potential for summation is contingent on understanding both convergence and the common ratio.