Problem 84 - 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. $$-6+5-\frac{25}{6}+\frac{125}{36}-\cdots$$

Step-by-Step Solution

Answer

The sum of the infinite geometric series is \(-\frac{36}{11}\)

Step 1: Identify The First Term and The Ratio

The first term of the series can be identified as \(-6\) and the ratio can be determined by dividing the second term by the first term, resulting to \(\frac{5}{-6}=-\frac{5}{6}\)

Step 2: Check If Sum Can Be Determined

The infinite geometric series would have a sum if the absolute value of the ratio is less than 1. The absolute value of \(-\frac{5}{6}\) is \(\frac{5}{6}\), which is less than 1; therefore, the sum can be determined.

Step 3: Apply Sum Formula

The formula for the sum of an infinite geometric series is \(S = \frac{a}{1 - r}\) where S is the sum, a is the first term, and r is the ratio. Substituting the values \(-6\) for a and \(-\frac{5}{6}\) for r, the sum can be calculated as \(S = \frac{-6}{1 - (-\frac{5}{6})} = \frac{-6}{1 + \frac{5}{6}} = \frac{-6}{\frac{11}{6}} = -\frac{36}{11}\)

Key Concepts

Geometric Progression

A geometric progression, or geometric sequence, is a series of numbers with a constant ratio between every pair of consecutive terms. This constant ratio is what makes each step between terms predictable and uniform. For instance, in the geometric series provided:

• The first term is \(-6\).
• The second term is \(5\).
• The third term is \(-\frac{25}{6}\).

If you observe these terms, you can see that each term is the product of the previous term and the constant term ratio. This ratio is crucial in determining the series' characteristics. It can be calculated by dividing any term by its preceding term, ensuring that the series is indeed geometric. Understanding this concept is fundamental as it lays the foundation for further calculations and interpretations of series-related mathematical ideas.

Convergence of Series

When dealing with infinite series, it is essential to understand whether the series converges or diverges. A series converges if its sum approaches a certain finite value as more terms are added. For a geometric series to converge, the absolute value of the common ratio must be less than 1. In simpler terms, the terms of the series must decrease in magnitude as they progress.

• If the absolute value of the ratio is less than 1, the terms get smaller and smaller, and the series converges.
• If the absolute value of the ratio is greater than or equal to 1, the terms do not shrink towards zero, causing the series to diverge.

In our example of the series \(-6 + 5 - \frac{25}{6} + \cdots\), the common ratio is \(-\frac{5}{6}\), and its absolute value is indeed less than 1. This confirms that the series converges, meaning we can find a finite sum for the entire series.

Sum Formula for Infinite Series

Once we've established that a geometric series converges, we can find the sum using the sum formula for infinite geometric series. The formula is given by:\[S = \frac{a}{1 - r} \]where \(S\) is the sum of the series, \(a\) is the first term, and \(r\) is the common ratio. To apply this in our scenario, we take:

• First term \(a = -6\).
• Common ratio \(r = -\frac{5}{6}\).

Plugging these into the formula, we calculate the sum:\[S = \frac{-6}{1 - (-\frac{5}{6})} = \frac{-6}{1 + \frac{5}{6}} = \frac{-6}{\frac{11}{6}} = -\frac{36}{11} \]This sum \-\frac{36}{11}\ is the finite sum of the infinite series, demonstrating the power and usefulness of the sum formula for converging geometric series. It allows mathematicians and students to easily and quickly find the sum of all terms in the series, rather than calculating each term individually and trying to sum them manually.