Work the following problems, showing de 1. Evaluate each of the following geometric series or state that it diverges: a. ΠΑΡΑ k (2 00 5 ΑΣΘ 12, k=0 b. Χ ΣΗ k=1 this sheet if you need more space. k C. ΣΟ k=0 312 k
Work the following problems, showing de 1. Evaluate each of the following geometric series or state that it diverges: a. ΠΑΡΑ k (2 00 5 ΑΣΘ 12, k=0 b. Χ ΣΗ k=1 this sheet if you need more space. k C. ΣΟ k=0 312 k
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
Evaluate each of the following geometric series or state that it diverges:
![The problem set provided asks students to evaluate each given geometric series or determine if it diverges. The specific problems and corresponding student annotations are as follows:
---
**Instructions:**
"Work the following problems, showing detailed work. Use the back of this sheet if you need more space."
**Evaluate each of the following geometric series or state that it diverges:**
---
**Problem a:**
\[
\sum_{k=0}^{\infty} \left( \frac{5}{12} \right)^k
\]
- Annotation: \( r = \frac{5}{12} < 1 \)
**Problem b:**
\[
\sum_{k=1}^{\infty} \left( -\frac{1}{3} \right)^k
\]
- Annotations include calculations involving limits showing:
\[
\lim_{k \to \infty} \left( -\frac{1}{3} \right)^k
\]
**Problem c:**
\[
\sum_{k=0}^{\infty} \left( \frac{3}{2} \right)^k
\]
- Annotations showing:
\[
\lim_{k \to \infty} \left( \frac{3}{2} \right)^k
\]
---
The notes in red indicate attempts or assessments made by the student, each marked with an "X," possibly indicating incorrect approaches or solutions. Each problem likely includes cross-references to the formula for the sum of a geometric series or evaluations involving the convergence criteria \( |r| < 1 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5ea198b5-e914-4142-850d-5851dbae1453%2F603369e1-bd13-4c36-acce-1a7cde639a94%2F9wxqwq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The problem set provided asks students to evaluate each given geometric series or determine if it diverges. The specific problems and corresponding student annotations are as follows:
---
**Instructions:**
"Work the following problems, showing detailed work. Use the back of this sheet if you need more space."
**Evaluate each of the following geometric series or state that it diverges:**
---
**Problem a:**
\[
\sum_{k=0}^{\infty} \left( \frac{5}{12} \right)^k
\]
- Annotation: \( r = \frac{5}{12} < 1 \)
**Problem b:**
\[
\sum_{k=1}^{\infty} \left( -\frac{1}{3} \right)^k
\]
- Annotations include calculations involving limits showing:
\[
\lim_{k \to \infty} \left( -\frac{1}{3} \right)^k
\]
**Problem c:**
\[
\sum_{k=0}^{\infty} \left( \frac{3}{2} \right)^k
\]
- Annotations showing:
\[
\lim_{k \to \infty} \left( \frac{3}{2} \right)^k
\]
---
The notes in red indicate attempts or assessments made by the student, each marked with an "X," possibly indicating incorrect approaches or solutions. Each problem likely includes cross-references to the formula for the sum of a geometric series or evaluations involving the convergence criteria \( |r| < 1 \).
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