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Chapter 10: Problem 42

Find the sum of each infinite geometric series. $$3-1+\frac{1}{3}-\frac{1}{9}+\cdots$$

Short Answer

Expert verified
The sum of the infinite geometric series is 2.25.

Step by step solution

01

Find the Common Ratio

Determine the ratio by dividing any term by its preceding term. For instance, \(-1/3 = -0.3333\) and \(1/(-1) = -1\). So, the common ratio, r, is -1/3.
02

Check if the Series Converges

A geometric series converges if the absolute value of r is less than 1. In our case, the absolute value of \(-1/3\) is \(1/3\), which is less than 1. Therefore, the given series does converge.
03

Apply the Sum Formula

In an infinite geometric series, the sum, S, is given by the formula \(S = a/(1-r)\), where a is the first term in the series, and r is the common ratio. Plugging the values, \(S = 3/(1-(-1/3)) = 3/(1+1/3) = 3/(4/3) = 9/4 = 2.25\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Ratio
The common ratio in a geometric series is crucial because it determines the relationship between consecutive terms. For any geometric series, the common ratio, denoted as \( r \), can be found by dividing any term in the series by the term immediately preceding it. It鈥檚 the factor by which you multiply one term to get to the next. In the series given in our exercise, the common ratio is derived from terms like \(-1\) and \(3\) such that \( \frac{-1}{3} = -\frac{1}{3} \).
  • Key Point: The common ratio provides insight into the progression of the series.
  • Example: If you have terms 3, -1, \( \frac{1}{3} \), then your common ratio will consistently show as \(-\frac{1}{3} \).
Knowing this, you can predict future values and behavior of the geometric series.
Convergence of Series
Determining if a geometric series converges is an essential step. Convergence refers to whether the series approaches a finite value as the number of terms extends to infinity. For a geometric series to converge, the absolute value of the common ratio must be less than one, \( |r| < 1 \). If a series diverges (opposite of converges), it means it keeps growing indefinitely and doesn't settle on any particular value.
  • Convergence Criterion: Absolute value of \( r \) should be less than one for convergence.
  • Example: If \( r = -\frac{1}{3} \), the series converges because \( \left| -\frac{1}{3} \right| = \frac{1}{3}\) which is less than 1.
Sum of Infinite Series
Once you've established that a geometric series converges, calculating the sum becomes the next logical step. The sum formula for an infinite geometric series \( S \) is expressed as \[S = \frac{a}{1 - r}\]where \( a \) is the first term of the series, and \( r \) is the common ratio. This formula allows you to precisely find the sum of an infinite sequence without needing to add up each term manually.
  • Application: For the series starting with 3 and having a common ratio of \(-\frac{1}{3}\), plug into the formula: \( S = \frac{3}{1 - (-\frac{1}{3})} \).
  • Result: It simplifies to \( S = \frac{9}{4} = 2.25 \).

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