4 4 th Find a formula for the n partial sum of the series 4 + 5 sum if the series converges. th The formula for the n" partial sum, sn, of the series is + -- 25 + 4 125 + ... + 5 4 + and use it to find the series'
4 4 th Find a formula for the n partial sum of the series 4 + 5 sum if the series converges. th The formula for the n" partial sum, sn, of the series is + -- 25 + 4 125 + ... + 5 4 + and use it to find the series'
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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![### Problem Statement
**Objective:**
Find a formula for the \( n^{th} \) partial sum of the series:
\[ 4 + \frac{4}{5} + \frac{4}{25} + \cdots + \frac{4}{5^{n-1}} + \cdots \]
and use it to find the series' sum if the series converges.
### Tasks
1. Determine the formula for the \( n^{th} \) partial sum, \( s_n \), of the series.
2. Explain how to find the sum of the series, assuming it converges.
### Solution Format
- The formula for the \( n^{th} \) partial sum, \( s_n \), of the series is given by:
\[ \boxed{\text{[Place Formula Here]}} \]
### Note
- This is a geometric series with the first term \( a = 4 \) and common ratio \( r = \frac{1}{5} \).
- Use the formula for the sum of the first \( n \) terms of a geometric series:
\[ s_n = a \frac{1-r^n}{1-r} \]
- Determine convergence by examining if \( |r| < 1 \), and apply the sum formula for an infinite geometric series:
\[ S = \frac{a}{1-r} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87314332-e2f5-4784-8bd6-50bd771df639%2Fc0eec546-3d30-4d5b-9e24-20978749c7a7%2Fev9g6cw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**Objective:**
Find a formula for the \( n^{th} \) partial sum of the series:
\[ 4 + \frac{4}{5} + \frac{4}{25} + \cdots + \frac{4}{5^{n-1}} + \cdots \]
and use it to find the series' sum if the series converges.
### Tasks
1. Determine the formula for the \( n^{th} \) partial sum, \( s_n \), of the series.
2. Explain how to find the sum of the series, assuming it converges.
### Solution Format
- The formula for the \( n^{th} \) partial sum, \( s_n \), of the series is given by:
\[ \boxed{\text{[Place Formula Here]}} \]
### Note
- This is a geometric series with the first term \( a = 4 \) and common ratio \( r = \frac{1}{5} \).
- Use the formula for the sum of the first \( n \) terms of a geometric series:
\[ s_n = a \frac{1-r^n}{1-r} \]
- Determine convergence by examining if \( |r| < 1 \), and apply the sum formula for an infinite geometric series:
\[ S = \frac{a}{1-r} \]
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