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Chapter 7: Q. 87 (page 605)
Let be a sequence such that and . Prove that role="math" localid="1649341558588" . (What you will have proven is that if the even terms and odd terms of a sequence both converge to L, then the sequence converges to L.)
Proved that
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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
If a positive finite number, what may we conclude about the two series?
Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.
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 .)
Determine whether the series converges or diverges. Give the sum of the convergent series.
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