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Chapter 7: Q. 18 (page 624)
Use the divergence test to analyze the given series. Each answer should either be the series diverges or the divergence test fails, along with the reason for your answer.
The divergence test failed as .
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If and diverges, explain why we cannot draw any conclusions about the behavior of.
Express each of the repeating decimals in Exercises 71鈥�78 as a geometric series and as the quotient of two integers reduced to lowest terms.
Use either the divergence test or the integral test to determine whether the series in Exercises 32鈥�43 converge or diverge. Explain why the series meets the hypotheses of the test you select.
35.
Letand be two convergent geometric series. If b and v are both nonzero, prove that is a geometric series. What condition(s) must be met for this series to converge?
Prove Theorem 7.31. That is, show that if a function a is continuous, positive, and decreasing, and if the improper integral converges, then the nth remainder, , for the series is bounded by
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