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Chapter 11: Problem 4

Determine the sums of the following geometric series when they are convergent. $$1+\frac{1}{2^{3}}+\frac{1}{2^{6}}+\frac{1}{2^{9}}+\frac{1}{2^{12}}+\cdots$$

Short Answer

Expert verified
The sum of the series is \( \frac{8}{7} \).

Step by step solution

01

- Identify the series

Recognize that the given series is a geometric series: \[1 + \frac{1}{2^{3}} + \frac{1}{2^{6}} + \frac{1}{2^{9}} + \frac{1}{2^{12}} + \text{...} \]
02

- Find the first term (a)

The first term of the series, denoted as \(a\), is 1.
03

- Determine the common ratio (r)

The common ratio \(r\) of the geometric series is the ratio between consecutive terms. Here, \[ r = \frac{\frac{1}{2^3}}{1} = \frac{1}{2^3} = \frac{1}{8}.\]
04

- Check for convergence

For a geometric series to be convergent, the common ratio \(|r|\) must be less than 1. Here, \[ |r| = \frac{1}{8} < 1, \] so the series is convergent.
05

- Use the formula for the sum of a convergent geometric series

The sum \(S\) of an infinite convergent geometric series is given by \[ S = \frac{a}{1 - r}. \]
06

- Apply the formula

Substitute values for \(a\) and \(r\): \[ S = \frac{1}{1 - \frac{1}{8}} = \frac{1}{\frac{7}{8}}. \]
07

- Simplify the expression

Simplify the fraction: \[ \frac{1}{\frac{7}{8}} = \frac{8}{7}. \]

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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 common ratio. The series can extend to infinity. For example, in the series: \[1 + \frac{1}{2^3} + \frac{1}{2^6} + \frac{1}{2^9} + \cdots\] each term is obtained by multiplying the previous term by \(\frac{1}{8}\).
Geometric series have these key properties:
  • The first term is usually denoted as \(a\).
  • The common ratio is represented by \(r\).
The terms grow or shrink exponentially depending on the value of \(r\). If \(|r| < 1\), the terms decrease in magnitude and the series may converge. Conversely, if \(|r| \geq 1\), the series typically diverges.
convergence criteria
To determine whether a geometric series converges, we must look at its common ratio, \(r\). For an infinite geometric series to converge, the absolute value of the common ratio must be less than 1: \(|r| < 1\).
This means the terms will get smaller and smaller, approaching zero. If the common ratio \(|r|\) is greater than or equal to 1, the series will diverge, meaning it does not settle on a finite sum.
Let's apply this to our example series: \[\frac{1}{2^3}, \frac{1}{2^6}, \frac{1}{2^9}, \cdots\] has a common ratio of \(\frac{1}{8}\).Since \(\left| \frac{1}{8} \right| < 1\), the series converges.
sum formula
Once we know that a geometric series converges, we can use a specific formula to find its sum. The sum, \(S\), of an infinite convergent geometric series is given by: \[ S = \frac{a}{1 - r} \] where \(a\) is the first term and \(r\) is the common ratio.
In our series: \(1, \frac{1}{2^3}, \frac{1}{2^6}, \cdots\), we first find that \(a = 1\) and \(r = \frac{1}{8}\). Using the sum formula, we can calculate: \[ S = \frac{1}{1 - \frac{1}{8}} = \frac{1}{\frac{7}{8}} = \frac{8}{7} \] Which simplifies to \(\frac{8}{7}\). This is the sum of our geometric series when it is convergent.

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