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
![**Transcription for Educational Website:**
---
**Evaluate:**
\[
\sum_{k=3}^{\infty} \frac{1}{(k-2)(k-1)}
\]
---
**Evaluate:**
\[
\sum_{k=2}^{\infty} \left( \frac{1}{3^k} - \frac{2}{3^{k+2}} \right)
\]
---
**Explanation:**
1. **First Expression:** The first expression is an infinite series starting from \( k=3 \) to infinity. The general term of the series is given by the fraction whose numerator is 1, and the denominator is the product of two linear terms \((k-2)\) and \((k-1)\).
2. **Second Expression:** Similarly, the second expression is an infinite series starting from \( k=2 \) to infinity. The general term here is composed of two parts: \( \frac{1}{3^k} \) and \( \frac{2}{3^{k+2}} \). Note that the powers of 3 decrease in the second term relative to those in the first term, impacting the series' convergence.
These expressions can involve calculus, particularly techniques like partial fraction decomposition or geometric series summation, for finding their sums.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F135565b3-f69f-45e7-bb45-4192b3629324%2F98e74ccb-710b-4257-b83d-843fbcd432be%2Fnq218p_processed.png&w=3840&q=75)
Transcribed Image Text:**Transcription for Educational Website:**
---
**Evaluate:**
\[
\sum_{k=3}^{\infty} \frac{1}{(k-2)(k-1)}
\]
---
**Evaluate:**
\[
\sum_{k=2}^{\infty} \left( \frac{1}{3^k} - \frac{2}{3^{k+2}} \right)
\]
---
**Explanation:**
1. **First Expression:** The first expression is an infinite series starting from \( k=3 \) to infinity. The general term of the series is given by the fraction whose numerator is 1, and the denominator is the product of two linear terms \((k-2)\) and \((k-1)\).
2. **Second Expression:** Similarly, the second expression is an infinite series starting from \( k=2 \) to infinity. The general term here is composed of two parts: \( \frac{1}{3^k} \) and \( \frac{2}{3^{k+2}} \). Note that the powers of 3 decrease in the second term relative to those in the first term, impacting the series' convergence.
These expressions can involve calculus, particularly techniques like partial fraction decomposition or geometric series summation, for finding their sums.
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