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Chapter 7: Q. 15 (page 603)
Give examples of sequences satisfying the given conditions or explain why such an example cannot exist.
A decreasing sequence that is bounded below but is not bounded above.
Examples of the sequences is .
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Provide a more general statement of the integral test in which the function f is continuous and eventually positive, and decreasing. Explain why your statement is valid.
Determine whether the series converges or diverges. Give the sum of the convergent series.
Express each of the repeating decimals in Exercises 71–78 as a geometric series and as the quotient of two integers reduced to lowest terms.
True/False:
Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.
(a) True or False: If , then converges.
(b) True or False: If converges, then .
(c) True or False: The improper integral converges if and only if the series converges.
(d) True or False: The harmonic series converges.
(e) True or False: If , the series converges.
(f) True or False: If as , then converges.
(g) True or False: If converges, then as .
(h) True or False: If and is the sequence of partial sums for the series, then the sequence of remainders converges to .
Express each of the repeating decimals in Exercises 71–78 as a geometric series and as the quotient of two integers reduced to lowest terms.
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