Compute the partial sums S3, S4, and Ss for the series and then find its sum. 00 Σ n+5 (Use symbolic notation and fractions where needed.) S3 = S4 = Ss = S =
Compute the partial sums S3, S4, and Ss for the series and then find its sum. 00 Σ n+5 (Use symbolic notation and fractions where needed.) S3 = S4 = Ss = S =
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:
Compute the partial sums \( S_3, S_4, \) and \( S_5 \) for the series and then find its sum.
\[
\sum_{n=1}^{\infty} \left( \frac{1}{n+4} - \frac{1}{n+5} \right)
\]
(Use symbolic notation and fractions where needed.)
### Partial Sums:
- \( S_3 = \) [Text Box]
- \( S_4 = \) [Text Box]
- \( S_5 = \) [Text Box]
### Sum:
- \( S = \) [Text Box]
### Explanation:
This exercise is about calculating the partial sums of a series given by the formula:
\[
\sum_{n=1}^{\infty} \left( \frac{1}{n+4} - \frac{1}{n+5} \right)
\]
Each partial sum \( S_N \) is computed by evaluating the series from \( n = 1 \) to \( n = N \) and summing the terms inside the expression \(\left(\frac{1}{n+4} - \frac{1}{n+5}\right)\).
- **\( S_3 \)**: Sum the first three terms of the series.
- **\( S_4 \)**: Sum the first four terms of the series.
- **\( S_5 \)**: Sum the first five terms of the series.
Finally, calculate the overall sum \( S \) of the infinite series.
Note: This type of series often telescopes, meaning most intermediate terms cancel out, simplifying the calculation of partial sums and making it easier to determine the sum.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8a52098d-a98f-4a3a-9ee3-9a8837591a20%2Fb981c2c0-d4c7-4bc4-aecc-8d5ce4d13866%2Foef5scu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem:
Compute the partial sums \( S_3, S_4, \) and \( S_5 \) for the series and then find its sum.
\[
\sum_{n=1}^{\infty} \left( \frac{1}{n+4} - \frac{1}{n+5} \right)
\]
(Use symbolic notation and fractions where needed.)
### Partial Sums:
- \( S_3 = \) [Text Box]
- \( S_4 = \) [Text Box]
- \( S_5 = \) [Text Box]
### Sum:
- \( S = \) [Text Box]
### Explanation:
This exercise is about calculating the partial sums of a series given by the formula:
\[
\sum_{n=1}^{\infty} \left( \frac{1}{n+4} - \frac{1}{n+5} \right)
\]
Each partial sum \( S_N \) is computed by evaluating the series from \( n = 1 \) to \( n = N \) and summing the terms inside the expression \(\left(\frac{1}{n+4} - \frac{1}{n+5}\right)\).
- **\( S_3 \)**: Sum the first three terms of the series.
- **\( S_4 \)**: Sum the first four terms of the series.
- **\( S_5 \)**: Sum the first five terms of the series.
Finally, calculate the overall sum \( S \) of the infinite series.
Note: This type of series often telescopes, meaning most intermediate terms cancel out, simplifying the calculation of partial sums and making it easier to determine the sum.
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