Problem 74 - 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. $$\sum_{n=1}^{\infty} 8\left(\frac{5}{3}\right)^{n-1}$$

Step-by-Step Solution

Answer

The given infinite geometric series does not sum to a finite value because the ratio exceeds 1.

Step 1: Identification of the ratio

Identify the ratio of the geometric series. Here the ratio \( r \) is \( \frac{5}{3} \)

Step 2: Applying condition for convergence

A geometric series will converge or sum to a finite value if and only if \( -1 < r < 1 \). Compare our ratio \( r \) with this.

Step 3: Final evaluation

Considering \( r = \frac{5}{3} \), it's evident that it is not within the range of \( -1 < r < 1 \), given that \( \frac{5}{3} > 1 \). Therefore, the given series does not sum to a finite value.

Key Concepts

Convergence of Series

When we discuss the convergence of series, particularly the infinite geometric series, it is crucial to understand when a series "converges." Convergence refers to whether the series approaches a specific value as you add more and more of its terms.
For an infinite geometric series, convergence depends on the ratio of consecutive terms, known as the geometric series ratio. This ratio must meet the criterion of being between -1 and 1 for the series to converge.

• If the series converges, it will sum to a finite number, meaning as you continue adding more and more terms, they will approach a certain value without ever surpassing it.
• If the series does not converge, the sum will grow indefinitely without stabilizing at any particular value.

When examining such series, the main indicator of convergence is the value of the common ratio.

Geometric Series Ratio

The geometric series ratio, often denoted as \( r \), is a fundamental element in understanding whether an infinite geometric series will converge. The common ratio \( r \) is obtained by dividing any term in the series by the preceding term.
For instance, in the exercise provided, the common ratio is \( r = \frac{5}{3} \). This ratio is uniform throughout the series and is crucial to determining convergence.

• The general rule is that if \( |r| < 1 \), the series converges.
• If \( |r| \geq 1 \), the series does not converge.

These conditions are essential because they dictate whether the sum will veer towards a finite number or grow without bounds. Understanding \( r \) gives us immediate insight into the behavior of the series.

Infinite Series Sum

Calculating the sum of an infinite geometric series is only possible when the series converges. Therefore, the sum of an infinite series is tied directly to the concept of convergence.
If \( |r| < 1 \), you can calculate the sum using the formula:
\[ S = \frac{a}{1 - r} \]
where \( S \) is the sum and \( a \) is the first term of the series. This formula yields a finite number that the series approaches as more terms are added. The beauty of this formula is how it transforms an infinite process into something finite and tangible.
However, in cases where \( |r| \geq 1 \), as in the example with \( r = \frac{5}{3} \), the series does not sum to a finite value. Instead, it diverges, meaning no finite sum can be achieved as you add more terms. Understanding this distinction is key to mastering the concepts of infinite series.