Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. (If the series is divergent, enter DIVERGENT.) -1+ 1+g-. X
Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. (If the series is divergent, enter DIVERGENT.) -1+ 1+g-. X
Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter13: Sequences And Series
Section13.CR: Chapter Review
Problem 6CC
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![**Determining Convergence or Divergence of an Infinite Geometric Series**
*Task*: Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. (If the series is divergent, enter "DIVERGENT.")
Given Series:
\[ \frac{81}{4} - \frac{9}{2} + 1 - \frac{2}{9} + \cdots \]
To enter your answer, utilize the input box provided:
\[ \boxed{} \]
An 'X' symbol indicates the input area for the answer.
**Instructions for Students**:
1. Identify the first term (\(a\)) of the geometric series.
2. Determine the common ratio (\(r\)).
3. Check if the absolute value of the common ratio is less than 1:
- If \(|r| < 1\), the series is convergent.
- Otherwise, if \(|r| \geq 1\), the series is divergent.
4. If the series is convergent, use the formula for the sum of an infinite geometric series:
\[ S = \frac{a}{1 - r} \]
5. Enter "DIVERGENT" if the series is divergent, or enter the sum if the series is convergent.
**Example Solution**:
Given the series:
\[ \frac{81}{4} - \frac{9}{2} + 1 - \frac{2}{9} + \cdots \]
First, identify the first term \(a = \frac{81}{4}\).
Next, determine the common ratio \(r\) by dividing the second term by the first term:
\[ r = \frac{-\frac{9}{2}}{\frac{81}{4}} = \cdots \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3f29774d-39d7-4c90-90ba-0f0924899c12%2Fa38af9d4-02f3-4b7b-ad2d-78463bae5124%2Fgsnkbzf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Determining Convergence or Divergence of an Infinite Geometric Series**
*Task*: Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. (If the series is divergent, enter "DIVERGENT.")
Given Series:
\[ \frac{81}{4} - \frac{9}{2} + 1 - \frac{2}{9} + \cdots \]
To enter your answer, utilize the input box provided:
\[ \boxed{} \]
An 'X' symbol indicates the input area for the answer.
**Instructions for Students**:
1. Identify the first term (\(a\)) of the geometric series.
2. Determine the common ratio (\(r\)).
3. Check if the absolute value of the common ratio is less than 1:
- If \(|r| < 1\), the series is convergent.
- Otherwise, if \(|r| \geq 1\), the series is divergent.
4. If the series is convergent, use the formula for the sum of an infinite geometric series:
\[ S = \frac{a}{1 - r} \]
5. Enter "DIVERGENT" if the series is divergent, or enter the sum if the series is convergent.
**Example Solution**:
Given the series:
\[ \frac{81}{4} - \frac{9}{2} + 1 - \frac{2}{9} + \cdots \]
First, identify the first term \(a = \frac{81}{4}\).
Next, determine the common ratio \(r\) by dividing the second term by the first term:
\[ r = \frac{-\frac{9}{2}}{\frac{81}{4}} = \cdots \]
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