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Chapter 7: Q. 58 (page 604)
Evaluate the limits in Exercises 55–60. Use the theorems in this section to justify each step of your work.
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Explain why a function a(x) has to be continuous in order for us to use the integral test to analyze a series for convergence.
Let be any real number. Show that there is a rearrangement of the terms of the alternating harmonic series that converges to . (Hint: Argue that if you add up some finite number of the terms of , the sum will be greater than . Then argue that, by adding in some other finite number of the terms of
, you can get the sum to be less than . By alternately adding terms from these two divergent series as described in the preceding two steps, explain why the sequence of partial sums you are constructing will converge to .)
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.
Let 0 < p < 1. Evaluate the limit
Explain why we cannot use a p-series with 0 < p < 1 in a limit comparison test to verify the divergence of the series
Explain why, if n is an integer greater than 1, the series diverges.
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