Chapter 7: Q.1 (page 630)
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: Iffor every and the improper integral role="math" localid="1651646409965" converges, then the improper integral .
(b) True or False: If for every x>0 and role="math" localid="1651646831120" , then the improper integrals both converge.
(c) True or False: If for every positive integer k, then the series converges.
(d) True and False: If for every positive integer k, then the series diverges.
(e) True or False: If for every positive integer k and the series converges, then the series converges.
(f) True or False: If and both diverge, then diverges.
(g) True or False: If and are both positive for every positive integer k and , then and both converge.
(h) True or False: Ifandboth converge, thenis finite.
Short Answer
a) True
b) False
c) False
d) False
e) False
f) False
g) False
h) False