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Chapter 7: Q. 17 (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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Determine whether the series converges or diverges. Give the sum of the convergent series.
Let f(x) be a function that is continuous, positive, and decreasing on the interval such that role="math" localid="1649081384626" . What can the divergence test tell us about the series ?
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
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 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 .)
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