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Chapter 7: Q. 7 (page 639)
Explain how you could adapt the ratio test to analyze a seriesk in which the terms of the series are all negative.
Hence explained.
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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 ?
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 .)
Whenever a certain ball is dropped, it always rebounds to a height p% (0 < p < 100) of its original position. What is the total distance the ball travels before coming to rest when it is dropped from a height of h meters?
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 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 ?
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