Problem 20 - Thomas Calculus

Question

In Exercises \(15-22\) , determine if the geometric series converges or diverges. If a series converges, find its sum. $$\left(\frac{1}{3}\right)^{-2}-\left(\frac{1}{3}\right)^{-1}+1-\left(\frac{1}{3}\right)+\left(\frac{1}{3}\right)^{2}-\cdots$$

Step-by-Step Solution

Answer

The series converges, and its sum is \(\frac{27}{4}\).

Step 1: Identify the First Term

First, identify the first term of the series. Here, the first term is \(a = \left(\frac{1}{3}\right)^{-2}\). Calculate its value: \[a = \left(\frac{1}{3}\right)^{-2} = 3^2 = 9.\]

Step 2: Determine the Common Ratio

To find the common ratio \(r\), divide the second term by the first term: \[r = \frac{\left(\frac{1}{3}\right)^{-1}}{\left(\frac{1}{3}\right)^{-2}} = \frac{3}{9} = \frac{1}{3}.\] Note that the series alternates signs, so technically \(r = -\frac{1}{3}\).

Step 3: Check Convergence Criterion

A geometric series converges if the absolute value of the common ratio is less than 1. Check if \(|r| < 1\): \[|r| = |-\frac{1}{3}| = \frac{1}{3} < 1.\] Since \(\frac{1}{3} < 1\), the series converges.

Step 4: Calculate the Sum of the Convergent Series

For a convergent geometric series, the sum \(S\) is given by \(S = \frac{a}{1-r}\). Substitute the values: \[S = \frac{9}{1 - (-\frac{1}{3})} = \frac{9}{1 + \frac{1}{3}} = \frac{9}{\frac{4}{3}} = 9 \times \frac{3}{4} = \frac{27}{4}.\]

Key Concepts

Convergence of Series

Understanding whether a series converges or diverges is crucial in mathematics. Convergence in a series means that as we continue adding its terms, the total approaches a certain finite number. For this to happen with geometric series, the absolute value of the common ratio must be less than 1.
When this condition is met, the series will not grow indefinitely, instead it settles around a finite sum. In our case, the geometric series converges because the common ratio's absolute value \(|r| = \rac{1}{3}\) is indeed less than 1.
This means, as we sum more and more terms of the series, the total is creeping closer to a specific number, which we can calculate.

Sum of Geometric Series

Finding the sum of a convergent geometric series involves using a specific formula. For an infinite geometric series where the first term is \(a\) and the common ratio is \(r\), the sum \(S\) is calculated by the formula: \[S = \rac{a}{1-r}.\]
This formula only holds when the absolute value of \(r\) is less than 1, which is why checking convergence is so important. In this exercise, we found \(a = 9\) and \(r = -\frac{1}{3}.\)
Plugging into the formula gives us: \[S = \rac{9}{1 - (-\frac{1}{3})} = \rac{9}{\frac{4}{3}} = \rac{27}{4}.\]
This result shows the total you’d get if you kept adding terms in the series indefinitely.

Common Ratio

The common ratio in a geometric series is the factor you multiply by to get from one term to the next. It's an essential element since it dictates how the series behaves.
In the given series, we determined the common ratio by dividing one term by the preceding one: \[r = \rac{\left(\frac{1}{3}\right)^{-1}}{\left(\frac{1}{3}\right)^{-2}} = -\frac{1}{3}.\]
The negative sign indicates that the series is alternating, meaning the terms switch between positive and negative. Recognizing the common ratio helps you understand the nature of the series, and whether convergence to a finite sum is possible.

Alternating Series

An alternating series is one where the terms alternate in sign. Such a pattern can lead to interesting convergence properties because positive and negative terms might cancel each other out.
In the series we examined, after identifying the common ratio as negative, we discovered that it creates an alternating series.

• This means that the signs of the terms change from positive to negative, and vice versa.
• This sign alternation helps in achieving convergence despite the terms themselves potentially being large, as they can negate each other.

Understanding that a series is alternating is important as it provides insight into how the terms contribute to the overall sum.