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Chapter 7: Q. 9 (page 603)
In Exercises 4–11, give examples of sequences satisfying the given conditions or explain why such an example cannot exist.
Two divergent sequences and such that the sequence diverges.
The example of the sequence is and .
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Explain why, if n is an integer greater than 1, the series 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.
Letand be two convergent geometric series. If b and v are both nonzero, prove that is a geometric series. What condition(s) must be met for this series to converge?
The contrapositive: What is the contrapositive of the implication “If A, then B.”?
Find the contrapositives of the following implications:
If a quadrilateral is a square, then it is a rectangle.
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