Problem 101
Question
In Exercises 93 - 106, find the sum of the infinite geometric series. \( \sum_{n=0}^{\infty}-3\left(0.9\right)^n \)
Step-by-Step Solution
Verified Answer
The sum of the given infinite geometric series is -30.
1Step 1: Identify the first term and the common ratio
In the given infinite geometric series \( \sum_{n=0}^{\infty}-3\left(0.9\right)^n \) , the first term (a) is -3 and the common ratio (r) is 0.9.
2Step 2: Check the validity for the infinite sum
The sum of an infinite geometric series can be found with the formula \( S = \frac{a}{1-r} \) if the absolute value of r is less than 1. Here, |0.9| < 1, so the series converges and the formula can be used.
3Step 3: Calculate the sum of the series
Plug values of a and r into the formula to get the sum: \( S = \frac{-3}{1 - 0.9} = \frac{-3}{0.1} = -30 \).
Key Concepts
Geometric SeriesCommon RatioConvergence of SeriesSeries Summation Formula
Geometric Series
A geometric series is a series of numbers in which each term is found by multiplying the previous term by a fixed, non-zero number. This number is called the "common ratio." The sequence of numbers forms a geometric progression, making them predictable and defined. In the context of your exercise, the series is infinite, as indicated by the summation symbol from 0 to infinity, and starts with a term of
-3, with a common ratio of 0.9.
Geometric series hold particular importance in mathematics due to their predictable patterns and the ability to calculate their sum, even if they stretch to infinity. Examples of geometric series include summations found in financial calculations, biological growth processes, and computer algorithms.
Geometric series hold particular importance in mathematics due to their predictable patterns and the ability to calculate their sum, even if they stretch to infinity. Examples of geometric series include summations found in financial calculations, biological growth processes, and computer algorithms.
Common Ratio
The common ratio is the factor by which we multiply each term to get the next term in a geometric series. In the given exercise, the common ratio is 0.9.
This ratio is crucial because it determines the behavior of the series.
This ratio is crucial because it determines the behavior of the series.
- A common ratio greater than 1 will cause the series to diverge, meaning it has no finite sum.
- A common ratio less than 1 will make the series converge, allowing us to find a finite sum.
Convergence of Series
The convergence of a series refers to the condition under which an infinite series approaches a finite sum. For a geometric series, convergence is determined by the absolute value of the common ratio.
If the absolute value of the common ratio \( r \) is less than 1, the series converges. This means we can compute the sum of the series using the series summation formula. If the absolute value of \( r \) is 1 or greater, the series does not converge and so has no finite sum.
In the exercise's problem, because \(|0.9| < 1\), we know the series converges, leading us straight toward calculating its sum.
If the absolute value of the common ratio \( r \) is less than 1, the series converges. This means we can compute the sum of the series using the series summation formula. If the absolute value of \( r \) is 1 or greater, the series does not converge and so has no finite sum.
In the exercise's problem, because \(|0.9| < 1\), we know the series converges, leading us straight toward calculating its sum.
Series Summation Formula
To find the sum of an infinite geometric series that converges, we use the series summation formula: \[ S = \frac{a}{1 - r} \] where \( S \) represents the sum, \( a \) is the first term, and \( r \) is the common ratio. This formula works efficiently for infinite series where \( |r| < 1 \). It allows us to find the sum swiftly without needing to manually add up infinite terms.
In the exercise, by substituting the first term \( a = -3 \) and the common ratio \( r = 0.9 \) into the formula, we calculate:\[ S = \frac{-3}{1 - 0.9} = \frac{-3}{0.1} = -30 \]This result, -30, is the sum of the infinite geometric series proposed in the original problem.
In the exercise, by substituting the first term \( a = -3 \) and the common ratio \( r = 0.9 \) into the formula, we calculate:\[ S = \frac{-3}{1 - 0.9} = \frac{-3}{0.1} = -30 \]This result, -30, is the sum of the infinite geometric series proposed in the original problem.
Other exercises in this chapter
Problem 100
In Exercises 97-102, use a calculator to find the sum. \( \displaystyle \sum_{k=0}^{4}\frac{(-1)^k}{k!} \)
View solution Problem 101
In Exercises 101-104, prove the property for all integers \( r \) and \( n \) where \( 0 \le r \le n \). \( _nC_r = _nC_{n - r} \)
View solution Problem 101
In Exercises 97-102, use a calculator to find the sum. \( \displaystyle \sum_{n=0}^{25}\frac{1}{4^n} \)
View solution Problem 102
In Exercises 101-104, prove the property for all integers \( r \) and \( n \) where \( 0 \le r \le n \). \( _nC_0 - _nC_1 + _nC_2 - \cdots \pm _nC_n = 0 \)
View solution