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Chapter 9: Problem 77

Find the sum of the infinite geometric series. $$8+6+\frac{9}{2}+\frac{27}{8}+\cdots$$

Short Answer

Expert verified
The sum of the given infinite geometric series is 32.

Step by step solution

01

Find the Ratio

To calculate the ratio, choose any term in the sequence and divide it by the previous term. Here, using second term divided by first term gives us:\( r = \frac{6}{8} = 0.75 \)
02

Use the Sum Formula

Now use the formula for the sum of an infinite geometric series, which is \( S = \frac{a}{1 - r} \). Here, a is the first term of the sequence, so a = 8 and r = 0.75 from our previous calculation. Substituting these values into the formula gives us:\( S = \frac{8}{1 - 0.75} = \frac{8}{0.25} = 32 \)

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Key Concepts

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

Geometric Series
A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the ratio. For example, in the series 8, 6, 4.5, ..., each term is 0.75 times the term before it. This pattern of multiplication is consistent throughout the series.

Understanding a geometric series is fundamental in many areas of mathematics and science. It can be finite with a set number of terms or infinite, extending indefinitely. In our example, we have an infinite series because the terms keep getting smaller but never reach zero. Geometric series can model many real-world scenarios, such as population growth, financial investments, and certain physical phenomena.
Sum Formula
The sum formula for an infinite geometric series can be used when the series is convergent; that is, when the absolute value of the common ratio is less than 1. This formula is given by the expression S = \( \frac{a}{1 - r} \), where S represents the sum of the series, a is the first term, and r is the common ratio.

For instance, to find the sum of 8+6+4.5+..., we identify the first term () as 8 and the common ratio (r) as 0.75. By applying these values to the sum formula, we find that S = \( \frac{8}{1 - 0.75} = 32 \). This is a powerful tool for summing infinitely many terms swiftly and elegantly.
Convergence of Series
For an infinite geometric series to be summable, it must converge, which means it approaches a finite value as more and more terms are added. Convergence occurs if the absolute value of the ratio, r, is less than 1. When |r| < 1, each successive term in the series becomes smaller, making the sum approach a specific value.

If the ratio were 1 or greater in absolute value, the terms would not diminish, which would lead to a series that grows without bound, known as a divergent series. Divergent series do not have a finite sum.

In our example, since 0.75 is less than 1, the series 8+6+4.5+... converges and we can use the sum formula to find that the infinite series sums to 32. Understanding convergence is critical to applying the correct methods for finding series sums and determines whether tasks in finance, physics, or other fields are feasible.

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