Problem 68 - Precalculus: Mathematics for Calculus

Question

Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. $$\frac{2}{5}+\frac{4}{25}+\frac{8}{125}+\cdots$$

Step-by-Step Solution

Answer

The series is convergent, and its sum is \(\frac{2}{3}\).

Step 1: Identify the first term

The first term of the series, denoted as \(a\), is the first number in the sequence. For this series, \(a = \frac{2}{5}\).

Step 2: Identify the common ratio

The common ratio \(r\) is found by dividing the second term by the first term. Calculate \(r = \frac{4/25}{2/5} = \frac{4}{25} \cdot \frac{5}{2} = \frac{20}{50} = \frac{2}{5}\).

Step 3: Determine convergence or divergence

A geometric series converges if the absolute value of the common ratio \(|r|\) is less than 1. Since \(|\frac{2}{5}| = 0.4 < 1\), the series is convergent.

Step 4: Calculate the sum of the series

The formula for the sum of an infinite convergent geometric series is \(S = \frac{a}{1 - r}\). Substituting the values, \(S = \frac{2/5}{1 - 2/5} = \frac{2/5}{3/5} = \frac{2}{3}\). Thus, the sum of the series is \(\frac{2}{3}\).

Key Concepts

Convergence and Divergence

In the world of series, understanding whether a series converges or diverges is crucial. For an infinite geometric series, this depends solely on the common ratio, denoted as \(r\). If the absolute value of \(r\), represented as \(|r|\), is less than 1, the series converges. A convergent series implies that as you add more terms, the sum approaches a specific finite number.
However, if \(|r|\) is equal to or greater than 1, the series diverges, meaning the sum grows indefinitely without approaching any limit. In other words, it just keeps getting larger or oscillates infinitely.
This fundamental concept serves as a basic check: always determine the common ratio first and assess its absolute value to conclude about convergence or divergence.

Sum of Infinite Series

Once you've established that a geometric series converges, computing its sum becomes straightforward. For a converging geometric series, the sum can be calculated using the formula: \[ S = \frac{a}{1 - r} \] where \(a\) is the first term and \(r\) is the common ratio.
Given that \(|r| < 1\), the denominator \(1-r\) will ensure the division results in a real, finite sum.
For example, in the series \(\frac{2}{5} + \frac{4}{25} + \frac{8}{125} + \cdots\), we've already determined that it converges because \(\left| \frac{2}{5} \right| = 0.4 < 1\). Substituting into the sum formula gives: - \(a = \frac{2}{5}\) - \(r = \frac{2}{5}\) - Thus, \( S = \frac{2/5}{1 - 2/5} = \frac{2/3} \) This result indicates that even though an infinite number of terms are added, the overall total will settle at \(\frac{2}{3}\).

Common Ratio

The common ratio is a key element of a geometric series development and understanding. It dictates the entire behavior of the series as it progresses term by term. To find the common ratio \(r\), you divide any term in the series by its preceding term.
This calculation reveals the factor by which each term multiplies to produce the following term. This ratio remains constant throughout the series.
For instance, in the geometric series \(\frac{2}{5} + \frac{4}{25} + \frac{8}{125} + \cdots\), measuring the common ratio involves taking \(\frac{4}{25} \div \frac{2}{5}\) to give \(\frac{2}{5}\).
The common ratio not only tells us about the size and sign of the growth or decay but also crucially determines whether the series converges or diverges. Remember, if \(|r| < 1\), the series will converge to a sum.