Chapter 7: Q. 5 (page 655)
Dominance Relationships for Sequences: Order the following sequences by dominance when
The required order of dominance sequence is
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Use either the divergence test or the integral test to determine whether the series in Given Exercises converge or diverge. Explain why the series meets the hypotheses of the test you select.
Determine whether the series converges or diverges. Give the sum of the convergent series.
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.
Let f(x) be a function that is continuous, positive, and decreasing on the interval such that , What can the integral tells us about the series ?
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