Problem 69 - Precalculus: Mathematics for Calculus

Question

Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. $$1+\frac{3}{2}+\left(\frac{3}{2}\right)^{2}+\left(\frac{3}{2}\right)^{3}+\cdots$$

Step-by-Step Solution

Answer

The series is divergent.

Step 1: Identify the first term and the common ratio

The given series is an infinite geometric series. Its first term is \( a = 1 \). The common ratio \( r \) can be found by dividing the second term by the first term: \( r = \frac{3}{2} \).

Step 2: Check for convergence

An infinite geometric series converges if its common ratio \( r \) satisfies \( |r| < 1 \). In this case, the common ratio \( r = \frac{3}{2} \) and \(|r| = \frac{3}{2} > 1\). Since the absolute value of the common ratio is greater than 1, the series is divergent.

Key Concepts

convergence

In the context of a geometric series, convergence refers to whether the series adds up to a finite value. For an infinite geometric series, this occurs only when the absolute value of the common ratio, denoted as \( r \), is less than 1. When \( |r| < 1 \), the terms get progressively smaller, approaching zero, and the series converges to a specific sum. This sum can be calculated using the formula for the sum of an infinite geometric series:

• \( S = \frac{a}{1-r} \)

where \( a \) is the first term of the series. Understanding convergence is crucial because it tells us under which conditions the infinite series will result in a tangible number instead of diverging off to infinity.

divergence

Divergence in an infinite geometric series means that the series does not sum up to a finite value. This happens when the absolute value of the common ratio, \( r \), is equal to or greater than 1. In these cases, the series terms either stay the same or increase over time instead of decreasing. As a result, the series grows indefinitely. For the series given in the original exercise, with a common ratio \( r = \frac{3}{2} \), we see that \(|r| = \frac{3}{2} > 1\), confirming its divergence. This means that we cannot find a sum for the series because it does not converge.

infinite series

An infinite series is essentially a sum of an infinite sequence of terms. Infinite series can be either geometric or non-geometric, and understanding their behavior is key in diverse fields like mathematics and physics. Geometric series, which are the focus here, have a constant ratio between successive terms. They are written as:

• \( a, ar, ar^2, ar^3, \ldots \)

It's important to evaluate whether an infinite series converges or diverges to understand its behavior, as this influences the capacity to calculate an actual sum or understand its growth over time. For example, financial models often use convergent geometric series for calculations, because they provide real, finite results.

common ratio

The common ratio in a geometric series is the ratio between any term and its preceding term. For an infinite geometric series, the common ratio \( r \) is a critical element, as it determines the behavior of the series over the infinite sequence. You calculate \( r \) by dividing any term in the series by the term right before it.

• Example: If your series is \( 1, \frac{3}{2}, \left(\frac{3}{2}\right)^2, \ldots \), the common ratio \( r = \frac{3}{2} \)

The magnitude of this ratio helps determine whether the series converges to a sum or diverges, as seen when \(|r| < 1\) leads to convergence and \(|r| \geq 1\) leads to divergence. A precise understanding of common ratio helps to predict the long-term behavior of the series.