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Chapter 10: Problem 26

Geometric series Evaluate each geometric series or state that it diverges. $$1+\frac{2}{7}+\frac{2^{2}}{7^{2}}+\frac{2^{3}}{7^{3}}+\cdots$$

Short Answer

Expert verified
$$ Answer: The sum of the given infinite geometric series is $$\frac{7}{5}.$$

Step by step solution

01

Determine the Common Ratio

Divide consecutive terms to find the common ratio (r): $$r = \frac{\frac{2^{n+1}}{7^{n+1}}}{\frac{2^n}{7^n}} = \frac{2}{7}$$
02

Check Convergence or Divergence

Determine if the series converges or diverges by checking if the common ratio is between -1 and 1: $$|r| = |\frac{2}{7}| < 1$$ Since the absolute value of the common ratio is less than 1, the series converges.
03

Find the Sum of the Series

Since the series converges, we can find the sum using the formula: $$S = \frac{a}{1-r}$$ The first term (a) is 1, and the common ratio (r) is 2/7: $$S = \frac{1}{1-\frac{2}{7}} = \frac{1}{\frac{5}{7}} = \frac{7}{5}$$ The sum of the geometric series is 7/5.

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Key Concepts

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

Common Ratio
In a geometric series, each term is obtained by multiplying the previous term by a fixed number, known as the common ratio. This ratio is crucial since it determines the behavior of the series. For our example \( 1 + \frac{2}{7} + \frac{2^2}{7^2} + \cdots \), you find the common ratio \( r \) by dividing any term by its preceding term. By doing so here, you find \( r = \frac{2}{7} \).
  • The common ratio is consistent across each term.
  • If it is constant, the series is geometric.
Understanding the common ratio helps you predict the series' behavior, and whether it will converge to a certain value or not.
Convergence
Convergence in geometric series occurs when the sum approaches a specific number as the series progresses indefinitely. For a geometric series to converge, its common ratio's absolute value must be less than 1. In our example, the common ratio \( |r| = |\frac{2}{7}| \). This value is less than 1, indicating that the series converges.
  • If \(|r| < 1\), the series converges.
  • If \(|r| \geq 1\), the series diverges.
Convergence ensures that the addition of infinite terms results in a finite sum, making it a crucial aspect of assessing geometric series.
Series Sum
The sum of an infinite geometric series can be calculated when the series converges. If a geometric series has a first term \(a\) and a common ratio \(r\), its sum \(S\) can be found using the formula:\[S = \frac{a}{1-r}\]In our series, the first term \(a\) is 1, and \(r\) is \(\frac{2}{7}\). Therefore, the sum is:\[S = \frac{1}{1-\frac{2}{7}} = \frac{1}{\frac{5}{7}} = \frac{7}{5}\]
  • This formula only works for convergent series.
  • The sum gives a single number to represent the infinite series.
Knowing how to find the series sum allows for practical applications of geometric series in solving real-world problems.
Divergence
Divergence refers to a series where the sum does not settle to a number, meaning it cannot be added up to result in a finite value. For geometric series, divergence occurs when the absolute value of the common ratio is 1 or greater. In such cases, as you sum more terms, the total grows indefinitely.
  • Divergence means no specific sum exists.
  • It's identified by \(|r| \geq 1\).
Understanding divergence prevents misconceptions when evaluating whether an infinite series can produce a finite sum, which is important when interpreting the results of long calculations.

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Most popular questions from this chapter

The expression $$\begin{aligned} &1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+}}}}\\\ &\end{aligned}$$ where the process continues indefinitely, is called a continued fraction. a. Show that this expression can be built in steps using the recurrence relation \(a_{0}=1, a_{n+1}=1+\frac{1}{a_{n}}\) for \(n=0,1,2,3, \ldots\) Explain why the value of the expression can be interpreted as \(\lim _{n \rightarrow \infty} a_{n},\) provided the limit exists. b. Evaluate the first five terms of the sequence \(\left\\{a_{n}\right\\}\) c. Using computation and/or graphing, estimate the limit of the sequence. d. Assuming the limit exists, use the method of Example 5 to determine the limit exactly. Compare your estimate with \(\frac{1+\sqrt{5}}{2},\) a number known as the golden mean. e. Assuming the limit exists, use the same ideas to determine the value of $$\begin{aligned} &a+\frac{b}{a+\frac{b}{a+\frac{b}{a+\frac{b}{a+}}}}\\\ &\end{aligned}$$ where \(a\) and \(b\) are positive real numbers.

Repeated square roots Consider the sequence defined by \(a_{n+1}=\sqrt{2+a_{n}}, a_{0}=\sqrt{2},\) for \(n=0,1,2,3, \ldots\) a. Evaluate the first four terms of the sequence \(\left\\{a_{n}\right\\} .\) State the exact values first, and then the approximate values. b. Show that the sequence is increasing and bounded. c. Assuming the limit exists, use the method of Example 5 to determine the limit exactly.

Determine whether the following series converge. Justify your answers. $$\sum_{k=1}^{\infty} 7^{-2 k}$$

Determine whether the following series converge. Justify your answers. $$\sum_{k=1}^{\infty} \frac{7^{k}+11^{k}}{11^{k}}$$

Determine whether the following series converge. Justify your answers. $$\sum_{k=1}^{\infty} \frac{4^{k^{2}}}{k !}$$
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