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8. Suppose (an-1 converges to A, and define a new sequence (b)-1 by bn = all n. Prove that (bn)n-1 converges to A. 1.00 (2 Im an +an+1 2 for

8. Suppose (an-1 converges to A, and define a new sequence (b)-1 by bn = all n. Prove that (bn)n-1 converges to A. 1.00 (2 Im an +an+1 2 for

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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#8 please? Thanks

8. Suppose (an) 1 converges to A, and define a new sequence (b)-1 by bn = an + an+1 2 80 n=1 for all n. Prove that (bn)-1 converges to A. converges to A, *9. Suppose (an)n-1, (bn)-1, and (c) are sequences such that lan (bn)=1 converges to A, and an ≤ c ≤ b, for all n. Prove that (c)-1 converges to A. *10. Prove that, if (an) converges to A, then {|an|1=1 converges to A. Is the converse true" Justify your conclusion. *11. Let (an)=1 be a sequence such that there exist numbers a and N such that, for n ≥ M =a. Prove that (an), converges to a. 8
Transcribed Image Text:8. Suppose (an) 1 converges to A, and define a new sequence (b)-1 by bn = an + an+1 2 80 n=1 for all n. Prove that (bn)-1 converges to A. converges to A, *9. Suppose (an)n-1, (bn)-1, and (c) are sequences such that lan (bn)=1 converges to A, and an ≤ c ≤ b, for all n. Prove that (c)-1 converges to A. *10. Prove that, if (an) converges to A, then {|an|1=1 converges to A. Is the converse true" Justify your conclusion. *11. Let (an)=1 be a sequence such that there exist numbers a and N such that, for n ≥ M =a. Prove that (an), converges to a. 8
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