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Chapter 19: Problem 8

Find the sums of the given infinite geometric series. $$9+8.1+7.29+\cdots$$

Short Answer

Expert verified
The sum of the infinite geometric series is 90.

Step by step solution

01

Identify the Series Parameters

The series given is an infinite geometric series of the form \( a + ar + ar^2 + \cdots \). The first term (\( a \)) is 9. The second term is 8.1, which means the common ratio (\( r \)) can be found by dividing the second term by the first term: \( r = \frac{8.1}{9} \). The common ratio simplifies to \( r = 0.9 \).
02

Check the Convergence Condition

For an infinite geometric series to converge, the absolute value of the common ratio must be less than 1. In this series, \( r = 0.9 \), and since \( |0.9| < 1 \), the series converges and the sum can be calculated.
03

Apply the Formula for Infinite Geometric Series Sum

The formula for the sum \( S \) of an infinite geometric series is \( S = \frac{a}{1 - r} \). Substitute \( a = 9 \) and \( r = 0.9 \) into the formula to find the sum:\[ S = \frac{9}{1 - 0.9} = \frac{9}{0.1} = 90. \]

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Key Concepts

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

Convergence Condition
The convergence condition for an infinite geometric series is crucial to determine whether we can actually find its sum. An infinite geometric series will only converge if the absolute value of its common ratio \( r \) is less than one. This means that the terms in the series get progressively smaller and closer to zero as you add more of them. In the case of our series, the common ratio \( r \) is 0.9. Since \( |0.9| < 1 \), it satisfies the convergence condition. This allows us to calculate a finite sum for the series, even though it has an infinite number of terms. If \( |r| \) is equal to or greater than 1, the series would diverge, meaning it would not sum to a finite value.
Common Ratio
The common ratio is a key feature that defines any geometric series. It is the factor by which we multiply each term in the series to get the next term. In an infinite geometric series of the form \( a + ar + ar^2 + \cdots \), the common ratio \( r \) is obtained by dividing one term by its preceding term. For the series in our example, we found \( r \) by dividing 8.1 (the second term) by 9 (the first term), which gave us \( r = 0.9 \). Understanding the common ratio is vital as it also helps determine the series' convergence. A smaller common ratio means the terms decrease more rapidly, leading to a quicker convergence.
Sum of Series
Finding the sum of an infinite geometric series might sound like a paradox, but thanks to its properties, we can compute it using a simple formula. Once we know the series converges, we can use the formula for the sum of an infinite geometric series: \( S = \frac{a}{1 - r} \). Here, \( a \) is the first term of the series, and \( r \) is the common ratio. Substituting the values we have from the exercise \( a = 9 \) and \( r = 0.9 \), we get:
  • First, find \( 1 - r = 0.1 \).
  • Then, divide the first term by this result \( \frac{9}{0.1} \).
  • The sum \( S = 90 \).
This means that, even though there are infinitely many terms, their total sum equals 90. This amazing property is why geometric series are so significant in mathematics.

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Most popular questions from this chapter

Find the indicated quantities for the appropriate arithmetic sequence. The sequence of ships' bells is as follows: 12: 30 A.M. one bell is rung, and each half hour later one more bell is rung than the previous time until eight bells are rung. The sequence is then repeated starting at 4: 30 A.M., again until eight bells are rung. This pattern is followed throughout the day. How many bells are rung in one day?

Solve the given problems by use of the sum of an infinite geometric series. Find the sum of the terms of the infinite series \(1+2 x+3 x^{2}+4 x^{3}+\cdots\) for \(|x|<1 .\) (Hint: Use \(S-x S\) )

Find the first four terms of the indicated expansions by use of the binomial series. $$\frac{1}{\sqrt{1-x}}$$

Find the indicated quantities.Do the squares of the terms of a geometric sequence also form a geometric sequence? Explain.

Find the indicated quantities for the appropriate arithmetic sequence. A college graduate is offered two positions. A computer company offers an annual salary of \(\$ 42,000\) with a guaranteed annual raise of \(\$ 1200 .\) A marketing company offers an annual salary of \(\$ 44,000\) with a guaranteed annual raise of \(\$ 600 .\) Which company will pay more for the first six years of employment, and how much more?
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