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If (Un )neN and (Un)neN are sequences such that (Un)neN and (UnUn)neN converge, then (Un)neN converges. Select one: a. False, here is a counter-example: un = e¯¹ and Un = e² b. O c. d. e2n for all n E N. True, because (un Un)neN will be bounded as a converging sequence, and thus (Un)neN is bounded and converges. False, here is a counter-example: un = 1/n² and Un = n for all n E N. True, because the product of two converging sequences is converging.

If (Un )neN and (Un)neN are sequences such that (Un)neN and (UnUn)neN converge, then (Un)neN converges. Select one: a. False, here is a counter-example: un = e¯¹ and Un = e² b. O c. d. e2n for all n E N. True, because (un Un)neN will be bounded as a converging sequence, and thus (Un)neN is bounded and converges. False, here is a counter-example: un = 1/n² and Un = n for all n E N. True, because the product of two converging sequences is converging.

Advanced Engineering Mathematics
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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If (un)neN and (Un)neN are sequences such that (Un)nÊN and (un Un)neN converge, then (Un)neN converges. Select one: a. False, here is a counter-example: un = e¯ª and Un = e²″ for all n € N. O b. True, because (un Un)neN will be bounded as a converging sequence, and thus (Un)neN is bounded and converges. O c. False, here is a counter-example: un = 1/n² and Un = n for all n E N. O d. True, because the product of two converging sequences is converging.
Transcribed Image Text:If (un)neN and (Un)neN are sequences such that (Un)nÊN and (un Un)neN converge, then (Un)neN converges. Select one: a. False, here is a counter-example: un = e¯ª and Un = e²″ for all n € N. O b. True, because (un Un)neN will be bounded as a converging sequence, and thus (Un)neN is bounded and converges. O c. False, here is a counter-example: un = 1/n² and Un = n for all n E N. O d. True, because the product of two converging sequences is converging.
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