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
57.
![### Telescoping Series Problems
In the following exercises, you are tasked with finding a formula for the nth term of the sequence of partial sums \(\{S_n\}\) for each telescoping series. After determining the formula, evaluate \(\lim_{n \to \infty} S_n\) to find the series' value or determine if it diverges.
---
#### Decimal Representations
50. \[1.25 = 1.252525\ldots\]
51. \[0.456 = 0.456456456\ldots\]
52. \[5.1283 = 5.12838383\ldots\]
53. \[0.00952 = 0.00952952\ldots\]
---
#### Telescoping Series
54. \[\sum_{k=1}^{\infty} \left( \frac{1}{k+2} - \frac{1}{k+3} \right)\]
55. \[\sum_{k=1}^{\infty} \left( \frac{1}{k+1} - \frac{1}{k+2} \right)\]
56. \[\sum_{k=1}^{\infty} \frac{20}{25k^2 + 15k - 4}\]
57. \[\sum_{k=1}^{8} \frac{1}{(k+6)(k+7)}\]
58. \[\sum_{k=3}^{\infty} \frac{10}{4k^2 + 32k + 63}\]
59. \[\sum_{k=3}^{\infty} \frac{4}{(4k - 3)(4k + 1)}\]
60. \[\sum_{k=0}^{\infty} \frac{1}{(3k+1)(3k+4)}\]
61. \[\sum_{k=1}^{\infty} \ln \frac{k+1}{k}\]
62. \[\sum_{k=3}^{\infty} \frac{2}{(2k-1)(2k+1)}\]
63. \[\sum_{k=1}^{\infty} \frac{1}{(k+p)(k+p+1)}\quad \text{where } p \text](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdaf8d145-0d6b-4ad9-b2a2-dcfeaef22c49%2F1064c436-aa1a-44f0-96a6-020c6af36422%2F25w3im9.jpeg&w=3840&q=75)
Transcribed Image Text:### Telescoping Series Problems
In the following exercises, you are tasked with finding a formula for the nth term of the sequence of partial sums \(\{S_n\}\) for each telescoping series. After determining the formula, evaluate \(\lim_{n \to \infty} S_n\) to find the series' value or determine if it diverges.
---
#### Decimal Representations
50. \[1.25 = 1.252525\ldots\]
51. \[0.456 = 0.456456456\ldots\]
52. \[5.1283 = 5.12838383\ldots\]
53. \[0.00952 = 0.00952952\ldots\]
---
#### Telescoping Series
54. \[\sum_{k=1}^{\infty} \left( \frac{1}{k+2} - \frac{1}{k+3} \right)\]
55. \[\sum_{k=1}^{\infty} \left( \frac{1}{k+1} - \frac{1}{k+2} \right)\]
56. \[\sum_{k=1}^{\infty} \frac{20}{25k^2 + 15k - 4}\]
57. \[\sum_{k=1}^{8} \frac{1}{(k+6)(k+7)}\]
58. \[\sum_{k=3}^{\infty} \frac{10}{4k^2 + 32k + 63}\]
59. \[\sum_{k=3}^{\infty} \frac{4}{(4k - 3)(4k + 1)}\]
60. \[\sum_{k=0}^{\infty} \frac{1}{(3k+1)(3k+4)}\]
61. \[\sum_{k=1}^{\infty} \ln \frac{k+1}{k}\]
62. \[\sum_{k=3}^{\infty} \frac{2}{(2k-1)(2k+1)}\]
63. \[\sum_{k=1}^{\infty} \frac{1}{(k+p)(k+p+1)}\quad \text{where } p \text
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