If (an)n is a sequence of positive real numbers that converges to 0, then the series Σan/√n converges. a. False, here is a counter-example: an = 1/n. b. True, because (an)neN is bounded (since it converges) and thus an/√n →0 by ALT. c. True, because an/√n →0 by ALT and thus, by n-th term test, an/√n converges. d. False, here is a counter-example: an = 1/√√n.
If (an)n is a sequence of positive real numbers that converges to 0, then the series Σan/√n converges. a. False, here is a counter-example: an = 1/n. b. True, because (an)neN is bounded (since it converges) and thus an/√n →0 by ALT. c. True, because an/√n →0 by ALT and thus, by n-th term test, an/√n converges. d. False, here is a counter-example: an = 1/√√n.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:If (an)n is a sequence of positive real numbers that converges to 0, then the series an/√n converges.
a. False, here is a counter-example: an
1/n.
b. True, because (an)neN is bounded (since it converges) and thus an/√n →0 by ALT.
c. True, because an/√n →0 by ALT and thus, by n-th term test, Σan/√n converges.
d. False, here is a counter-example: an = 1/√√n.
=
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