Problem 119
Question
Exercises 117-120, find the sum of the infinite series. \( \displaystyle \sum_{k=1}^{\infty} 7 \left(\frac{1}{10} \right)^k \)
Step-by-Step Solution
Verified Answer
The sum of the infinite series is \( \frac{7}{9} \).
1Step 1: Identify the first term and the common ratio
The first term \(a\) is the value of the first term in the series, which is \(7(\frac{1}{10})^1 = 0.7\). The common ratio \(r\) is the factor that each term is multiplied by to get the next term, which is \(\frac{1}{10}\).
2Step 2: Apply the formula
Plug \(a\) and \(r\) into the formula. \(S = \frac{0.7}{1 - \frac{1}{10}}\)
3Step 3: Compute the result
Simplify the denominator to get \(1 - \frac{1}{10} = \frac{9}{10}\). Then compute \(S = \frac{0.7}{\frac{9}{10}} = \frac{7}{9}\).
Key Concepts
Sum of Infinite SeriesGeometric Series FormulaCommon Ratio
Sum of Infinite Series
An infinite series is a sequence of numbers that keeps going on indefinitely. A common task with these series is to find their sum, especially when the series converges, meaning it approaches a specific value. For an infinite geometric series, the sum can be determined using a specific formula as long as the series meets particular criteria.
An infinite geometric series takes the form: \[ a + ar + ar^2 + ar^3 + ext{...} \] where 'a' is the first term and 'r' is the common ratio. The sum of an infinite geometric series is expressed as: \[ S = \frac{a}{1 - r} \] This formula only works when the absolute value of the common ratio \( r \) is less than one, \(|r| < 1\). If \( |r| \) is equal to or greater than one, the series either diverges or does not converge to a particular number.
An infinite geometric series takes the form: \[ a + ar + ar^2 + ar^3 + ext{...} \] where 'a' is the first term and 'r' is the common ratio. The sum of an infinite geometric series is expressed as: \[ S = \frac{a}{1 - r} \] This formula only works when the absolute value of the common ratio \( r \) is less than one, \(|r| < 1\). If \( |r| \) is equal to or greater than one, the series either diverges or does not converge to a particular number.
Geometric Series Formula
The geometric series formula is a powerful tool used to find the sum of terms in a geometric sequence. A geometric sequence is one where each term after the first is found by multiplying the previous term by a fixed, non-zero number known as the common ratio.
The formula to find the sum \( S \) of a finite geometric series is given by: \[ S_n = a \frac{1 - r^n}{1 - r} \] Where \( S_n \) is the sum of the first 'n' terms, \( a \) is the first term, and \( r \) is the common ratio. However, for an infinite series, as seen in the previous section, the sum simplifies assuming \( |r| < 1 \), and becomes \( S = \frac{a}{1 - r} \).
The formula to find the sum \( S \) of a finite geometric series is given by: \[ S_n = a \frac{1 - r^n}{1 - r} \] Where \( S_n \) is the sum of the first 'n' terms, \( a \) is the first term, and \( r \) is the common ratio. However, for an infinite series, as seen in the previous section, the sum simplifies assuming \( |r| < 1 \), and becomes \( S = \frac{a}{1 - r} \).
- This simplification arises from the series tending towards a finite sum as increasing numbers of terms are added, as long as \( r \) is between -1 and 1, ensuring the series converges.
- This condition causes the terms to get smaller, effectively making their contribution to the sum negligible as the series progresses indefinitely.
Common Ratio
The common ratio is a fundamental concept in understanding geometric series and sequences. It defines how each term in the series relates to the previous one.
In a geometric series, the common ratio \( r \) is constant, meaning each term is derived by multiplying the previous term by this ratio. For example, in the series from the exercise, the common ratio \( r \) is \( \frac{1}{10} \). This implies each term is ten times smaller than the term before it.
In a geometric series, the common ratio \( r \) is constant, meaning each term is derived by multiplying the previous term by this ratio. For example, in the series from the exercise, the common ratio \( r \) is \( \frac{1}{10} \). This implies each term is ten times smaller than the term before it.
- To find \( r \), take any term from the series and divide it by the one before it.
- The series converges if \( |r| < 1 \), meaning the terms are getting smaller and closer to zero.
- This behavior makes it possible to sum an infinite number of terms to reach a finite value.
Other exercises in this chapter
Problem 118
A deposit of \(50 is made at the beginning of each month in an account that pays 8% interest, compounded monthly. The balance \) A \( in the account at the end
View solution Problem 118
Exercises 117-120, find the sum of the infinite series. \( \displaystyle \sum_{k=1}^{\infty} \left(\frac{1}{10} \right)^k \)
View solution Problem 120
A deposit \( P \) of dollars is made at the beginning of each month in an account with an annual interest rate \( r \) compounded continuously. The balance \( A
View solution Problem 120
Exercises 117-120, find the sum of the infinite series. \( \displaystyle \sum_{i=1}^{\infty} 2 \left(\frac{1}{10} \right)^i \)
View solution