Problem 81 - 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. $$3+\frac{15}{2}+\frac{75}{4}+\frac{375}{8}+\cdots$$

Step-by-Step Solution

Answer

The sum of the series cannot be found since the series does not converge. This is because the common ratio is 2.5, which is greater than 1.

Step 1: Identify first term and common ratio

Firstly, identify the first term \( a \) of the geometric series, which in this case is 3. Then, to identify the common ratio \( r \), divide the second term in the series by the first, so \( \frac{15}{2}/3 = \frac{15}{6} = 2.5 \). So, \( a = 3 \) and \( r = 2.5 \).

Step 2: Check for convergence

A geometric series can be summed to infinity only if the absolute value of the common ratio is less than 1. In this case, \( |r| = |2.5| = 2.5 \), which is greater than 1. Thus, this infinite series does not converge, meaning that its sum cannot be found.

Key Concepts

Sum of Geometric Series

Understanding the sum of a geometric series is crucial when dealing with patterns where each term is a constant multiple of the previous term. In mathematical terms, a geometric series takes the form of \(a + ar + ar^2 + ar^3 + \cdots\), where \(a\) is the first term and \(r\) is the common ratio. For an infinite geometric series to have a sum, a very specific condition must be met: the absolute value of the common ratio \(|r|\) must be less than one. When this condition holds, the sum \(S\) of the infinite series can be calculated using the formula \(S = \frac{a}{1 - r}\).

Convergence of Series

The convergence of a series is a concept that determines whether or not the series approaches a finite value as the number of terms increases indefinitely. In the case of a geometric series, this hinges on the size of the common ratio. When \(|r| < 1\), the series will converge because the terms get smaller and smaller, adding an increasingly negligible amount to the sum. However, if \(|r| \ge 1\), the terms do not diminish, and the series diverges, meaning it does not sum up to a finite number. Our example series did not meet this convergence criterion, given the common ratio of 2.5, leading to the conclusion that it diverges and does not have a defined sum.

Common Ratio

The common ratio in a geometric series plays a pivotal role in both the series' behavior and in finding its sum. It is the factor by which we multiply one term to get the next. Calculating it is straightforward: we divide any term by its preceding term. This ratio, as we've seen, also dictates whether the series is convergent or divergent. It's important to note that if the common ratio is negative, the terms will alternate in sign. While this does not inherently affect convergence, it does alter the series' behavior, leading to an oscillating pattern.

Geometric Sequence

A geometric sequence is simply the ordered set of terms involved in a geometric series, without the summation. It's a list of numbers where each term after the first is found by multiplying the previous term by the common ratio. An example of a geometric sequence is \(3, 7.5, 18.75, \cdots\), following the pattern where each term is a constant multiple of its predecessor. If you need to verify whether a set of numbers form a geometric sequence, just divide consecutive terms to see if the common ratio remains constant throughout.

When discussing geometric sequences, it is sometimes helpful to use the nth term formula: \(a_n = a \times r^{(n-1)}\) where \(a_n\) is the nth term of the sequence. This allows for quick computation of any term in the sequence without calculating all preceding terms.