Chapter 7: Q. 4 (page 652)
Find an example of a divergent series of the form
(a) that satisfies conditions (i) and (iii), but not condition (ii);
(b) that satisfies conditions (i) and (ii), but not condition (iii).
Short Answer
(i)
Chapter 7: Q. 4 (page 652)
Find an example of a divergent series of the form
(a) that satisfies conditions (i) and (iii), but not condition (ii);
(b) that satisfies conditions (i) and (ii), but not condition (iii).
(i)
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Get started for freeProve Theorem 7.25. That is, show that the series either both converge or both diverge. In addition, show that if converges to L, thenconverges tolocalid="1652718360109"
Determine whether the series converges or diverges. Give the sum of the convergent series.
Express each of the repeating decimals in Exercises 71–78 as a geometric series and as the quotient of two integers reduced to lowest terms.
True/False:
Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.
(a) True or False: If , then converges.
(b) True or False: If converges, then .
(c) True or False: The improper integral converges if and only if the series converges.
(d) True or False: The harmonic series converges.
(e) True or False: If , the series converges.
(f) True or False: If as , then converges.
(g) True or False: If converges, then as .
(h) True or False: If and is the sequence of partial sums for the series, then the sequence of remainders converges to .
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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