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Explain why or why not Determine whether the following state- ments are true and give an explanation or counterexample. a. The terms of the sequence {a,} increase in magnitude, so the limit of the sequence does not exist. b. The terms of the series E 1 approach zero, so the series converges. c. The terms of the sequence of partial sums of the series a, 5 approach , so the infinite series converges to 5 2 d. An alternating series that converges absolutely must converge conditionally. e. The sequence a, converges. n? + 1 (-1)"n² n? + 1 f. The sequence a, converges. k² g. The series > converges. k? + 1 k=1 h. The sequence of partial sums associated with the series 1 Σ converges. k2 + 1 k=1

Explain why or why not Determine whether the following state- ments are true and give an explanation or counterexample. a. The terms of the sequence {a,} increase in magnitude, so the limit of the sequence does not exist. b. The terms of the series E 1 approach zero, so the series converges. c. The terms of the sequence of partial sums of the series a, 5 approach , so the infinite series converges to 5 2 d. An alternating series that converges absolutely must converge conditionally. e. The sequence a, converges. n? + 1 (-1)"n² n? + 1 f. The sequence a, converges. k² g. The series > converges. k? + 1 k=1 h. The sequence of partial sums associated with the series 1 Σ converges. k2 + 1 k=1

Calculus: Early Transcendentals
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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Explain why or why not Determine whether the following state- ments are true and give an explanation or counterexample. a. The terms of the sequence {a,} increase in magnitude, so the limit of the sequence does not exist. b. The terms of the series E 1 approach zero, so the series converges. c. The terms of the sequence of partial sums of the series a, 5 approach , so the infinite series converges to 5 2 d. An alternating series that converges absolutely must converge conditionally. e. The sequence a, converges. n? + 1 (-1)"n² n? + 1 f. The sequence a, converges. k² g. The series > converges. k? + 1 k=1 h. The sequence of partial sums associated with the series 1 Σ converges. k2 + 1 k=1
Transcribed Image Text:Explain why or why not Determine whether the following state- ments are true and give an explanation or counterexample. a. The terms of the sequence {a,} increase in magnitude, so the limit of the sequence does not exist. b. The terms of the series E 1 approach zero, so the series converges. c. The terms of the sequence of partial sums of the series a, 5 approach , so the infinite series converges to 5 2 d. An alternating series that converges absolutely must converge conditionally. e. The sequence a, converges. n? + 1 (-1)"n² n? + 1 f. The sequence a, converges. k² g. The series > converges. k? + 1 k=1 h. The sequence of partial sums associated with the series 1 Σ converges. k2 + 1 k=1
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