and (bn) be two sequences of real number disprove each of the following statements: n=1 n=1 ¹) If the sequence (an) is defined by the re 1 = −2 − a², a₁ = 1, then (an)1 converges. - 81x -) If an bn for every n E N and lim an and =t, then lim (an-bn) 0. 81x -) If for every sequence of real numbers (c Cn does not exist, we have lim (an+cn) does an exists. 81x verges. If lim (a2n-an) = 0, then the sequence 81x
and (bn) be two sequences of real number disprove each of the following statements: n=1 n=1 ¹) If the sequence (an) is defined by the re 1 = −2 − a², a₁ = 1, then (an)1 converges. - 81x -) If an bn for every n E N and lim an and =t, then lim (an-bn) 0. 81x -) If for every sequence of real numbers (c Cn does not exist, we have lim (an+cn) does an exists. 81x verges. If lim (a2n-an) = 0, then the sequence 81x
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:5. Let (an) and (bn) be two sequences of real numbers.
Prove or disprove each of the following statements:
(a) (*. 7) If the sequence (an)n-1 is defined by the recursive formula
an+1 = -2-a, a₁ = 1, then (an)=1 converges.
(b) (^_ a) If an
exist, then lim
n4x
(c) {
-) If for every sequence of real numbers (Cn)1 for which
lim Cn does not exist, we have lim (an+cn) does not exist, then
818
lim an exists.
81x
(d) (-- .
bn for every n e N and lim an and lim bn do not
(an-bn) 0.
84x
84x
converges.
848
If lim (a2n-an) = 0, then the sequence (an)n=1
84x
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