Problem 83 - Algebra and Trigonometry Real Mathematics, Real People

Question

Finding the Sum of an Infinite Geometric Series Find the sum of the infinite geometric series, if possible. If not possible, explain why. $$-7+2-\frac{4}{7}+\frac{8}{49}-\cdots$$

Step-by-Step Solution

Answer

The sum of the infinite geometric series \(-7+2-\frac{4}{7}+\frac{8}{49}-\cdots\) is \(-\frac{49}{9}\).

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

Here, looking at the series, we identify the first term in the given series as -7 (a = -7). The common ratio r can be found by dividing the second term by the first term, or the third term by the second term, etc. So, if we take the second term (2) divided by the first term (-7), we get \(- \frac{2}{7}\). We verify this by also checking another set of successive terms. The third term is \(- \frac{4}{7}\), and when divided by 2, it also gives us - \(\frac{2}{7}\). Therefore, we identify the common ratio as - \(\frac{2}{7}\) (r = -\frac{2}{7}).

Step 2: Apply the infinite geometric series formula

We know the formula for the sum of an infinite geometric series is \[S = \frac{a}{1 - r}\]. Substituting -7 for a and \(\frac{-2}{7}\) for r, we get \[S = \frac{-7}{1 - \left(-\frac{2}{7}\right)}\]. Simplifying this, we get \[S = \frac{-7}{1 + \frac{2}{7}} = \frac{-7}{\frac{9}{7}} = -\frac{7}{\frac{9}{7}} = -\frac{49}{9}\].

Step 3: State the result

So, the sum of the given infinite geometric series is \(-\frac{49}{9}\).

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 constant called the common ratio. For example, in the series \(1, 2, 4, 8, 16, \ldots\), each term is twice the previous term, making the common ratio 2. In our exercise, the series -7, +2, -\(\frac{4}{7}\), +\(\frac{8}{49}\), \ldots follows a pattern where each term is multiplied by -\(\frac{2}{7}\) to get the next term.

A crucial aspect of geometric series is determining this common ratio, as it is instrumental in finding the sum of the series, especially when the series is infinite. If the common ratio is between -1 and 1 (excluding the endpoints), the infinite series converges, meaning it approaches a specific finite value as more terms are added. Our goal with the infinite geometric series is to find out what that sum is if it exists.

Convergence of Series

The convergence of a series is a fundamental concept in mathematics, particularly in calculus and analysis. It refers to the behavior of a series as the number of terms increases indefinitely. A series is said to converge if the partial sums of the series approach a specific limit. For the geometric series, this occurs when the common ratio has an absolute value less than 1. Conversely, if the absolute value of the common ratio is greater than or equal to 1, the series diverges, meaning its sum grows without bound or oscillates indefinitely.

In the provided exercise, the common ratio is -\(\frac{2}{7}\), which lies between -1 and 1. This means our series converges to a single value as more terms are added, making the task of finding the sum of the infinite series a feasible one. Identifying the nature of the series—whether it converges or diverges—is a crucial step before applying any formula to find a sum.

Series Formulas

To find the sum of a series, especially an infinite geometric series, we use a specific formula. The sum \(S\) of an infinite geometric series with the first term \(a\) and common ratio \(r\), provided \(|r|<1\), is given by \[S = \frac{a}{1 - r}\]. This elegant formula provides a shortcut to directly calculate the sum without having to add up an infinite number of terms. In our example, the series starts with \(-7\) and has a common ratio of -\(\frac{2}{7}\), which fits within the necessary bounds to use the formula.

When you substitute the identified values into the series formula, the calculation becomes straightforward. The sum is the numerator (the first term), divided by the difference between 1 and the common ratio. Through simplification, as shown in the step-by-step solution, you arrive at a single, finite number that represents the sum of an infinitely long list of numbers, showcasing the power of series formulas in mathematics.