Problem 1
Question
Find the sum of each infinite geometric series, if it exists. \(a_{1}=36, r=\frac{2}{3}\)
Step-by-Step Solution
Verified Answer
The sum of the series is 108.
1Step 1: Verify the conditions for sum of infinite geometric series
To find the sum of an infinite geometric series, we must first check if it meets the conditions for converging. The series converges if the absolute value of the common ratio \(r\) is less than 1, i.e., \(|r| < 1\). In this case, \(r = \frac{2}{3}\) and \(\left| \frac{2}{3} \right| = \frac{2}{3} < 1\). Therefore, the series converges.
2Step 2: Apply formula for sum of an infinite geometric series
For an infinite geometric series with first term \(a_1\) and common ratio \(r\), the sum \(S\) can be calculated using the formula: \[ S = \frac{a_1}{1 - r} \]Given \(a_1 = 36\) and \(r = \frac{2}{3}\), we substitute these values into the formula.
3Step 3: Substitute values and calculate the sum
Substitute \(a_1 = 36\) and \(r = \frac{2}{3}\) into the formula: \[ S = \frac{36}{1 - \frac{2}{3}} \]Compute the denominator: \[ 1 - \frac{2}{3} = \frac{1}{3} \]Now substitute back into the formula: \[ S = \frac{36}{\frac{1}{3}} \]To divide by a fraction, multiply by its reciprocal: \[ S = 36 \times 3 = 108 \]
4Step 4: Conclusion
The sum of the infinite geometric series with \(a_1 = 36\) and \(r = \frac{2}{3}\) is calculated as 108. Since \(|r| < 1\), the series converges, and our calculation is valid.
Key Concepts
Sum of a SeriesConvergenceGeometric Series Formula
Sum of a Series
Understanding the sum of a series is crucial in grasping the concept of infinite geometric series. When we say "sum of a series," we are referring to the total value obtained when all terms in a sequence are added together.
In a geometric series: - Each term after the first is obtained by multiplying the previous term by a constant called the "common ratio." - The series can be finite or infinite. In an infinite series, there are countless terms, and one might wonder how we could ever find a sum for something infinite. Luckily, under certain conditions, the sum does exist, and it is surprisingly simple to calculate. For a geometric series, if it meets the criteria of converging, we can use a specific formula to compute its sum, even if it has an infinite number of terms. This involves a clear understanding of convergence and the geometric series formula itself.
In a geometric series: - Each term after the first is obtained by multiplying the previous term by a constant called the "common ratio." - The series can be finite or infinite. In an infinite series, there are countless terms, and one might wonder how we could ever find a sum for something infinite. Luckily, under certain conditions, the sum does exist, and it is surprisingly simple to calculate. For a geometric series, if it meets the criteria of converging, we can use a specific formula to compute its sum, even if it has an infinite number of terms. This involves a clear understanding of convergence and the geometric series formula itself.
Convergence
One key criterion for finding the sum of an infinite geometric series is whether it converges. Simply put, convergence means that as we keep adding more terms in the series, the total sum gets closer and closer to a specific number.
To determine if a geometric series converges, check the common ratio, denoted as \(r\). The series converges if the absolute value of \(r\) is less than 1: - \( |r| < 1 \)
Why \( |r| < 1 \)? When the common ratio is a fraction (less than 1 but more than -1), each term in the series becomes smaller and smaller. This allows the terms' total sum to approach a fixed value, instead of growing indefinitely or oscillating.
If the common ratio \( |r| \) is 1 or more, the series does not converge, and thus, does not have a finite sum. Understanding this condition is crucial because it assures us that the series' sum, like magic, can indeed be finite, even though initially, it seems counterintuitive because it has infinite terms.
To determine if a geometric series converges, check the common ratio, denoted as \(r\). The series converges if the absolute value of \(r\) is less than 1: - \( |r| < 1 \)
Why \( |r| < 1 \)? When the common ratio is a fraction (less than 1 but more than -1), each term in the series becomes smaller and smaller. This allows the terms' total sum to approach a fixed value, instead of growing indefinitely or oscillating.
If the common ratio \( |r| \) is 1 or more, the series does not converge, and thus, does not have a finite sum. Understanding this condition is crucial because it assures us that the series' sum, like magic, can indeed be finite, even though initially, it seems counterintuitive because it has infinite terms.
Geometric Series Formula
The geometric series formula is a straightforward tool to find the sum of an infinite series, given the series meets the convergence condition of \(|r| < 1\).
The formula for the sum \(S\) is:\[ S = \frac{a_1}{1 - r} \] Where:
As seen in the example given, substituting \(a_1 = 36\) and \(r = \frac{2}{3}\) into the formula yields:
The formula for the sum \(S\) is:\[ S = \frac{a_1}{1 - r} \] Where:
- \(a_1\) is the first term of the series,
- \(r\) is the common ratio of the series.
As seen in the example given, substituting \(a_1 = 36\) and \(r = \frac{2}{3}\) into the formula yields:
- First calculate \(1 - r\), which is \(1 - \frac{2}{3} = \frac{1}{3}\),
- Then, the sum \(S = \frac{36}{\frac{1}{3}} = 108\).
Other exercises in this chapter
Problem 1
Expand each power. $$ (p+q)^{5} $$
View solution Problem 1
Find the first five terms of each sequence. $$ a_{1}=12, a_{n+1}=a_{n}-3 $$
View solution Problem 1
Find \(S_{n}\) for each geometric series described. $$ a_{1}=5, r=2, n=14 $$
View solution Problem 1
Find the next four terms of each arithmetic sequence. \(12,16,20, \dots\)
View solution