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trying to prove!). Suppose that 0 < an < bn for all n > 1, and define Sk = a1 + a2 + ... + ak. Suppose also that L= E bn (so the series converges to a sum L.) (a) Suppose that an > 0 for all n. How do we know that the sequence {S} is increasing? Show directly that Sk < Sk+1 (b) Explain why the sequence {Sk} is bounded. (c) Explain why {Sk} converges. What relevant Theorem is used to conclude this? (d) What does the fact that {Sk} converges have to do with convergence of En=1 an? (e) Would part a) be true if an values could be any real numbers (so positive or negative)? If you say yes, then support your claim with a brief argument. If you say no, then give a counterexample, i.e. an example of a sequence {a.,} for which {S.} is not increasing.

trying to prove!). Suppose that 0 < an < bn for all n > 1, and define Sk = a1 + a2 + ... + ak. Suppose also that L= E bn (so the series converges to a sum L.) (a) Suppose that an > 0 for all n. How do we know that the sequence {S} is increasing? Show directly that Sk < Sk+1 (b) Explain why the sequence {Sk} is bounded. (c) Explain why {Sk} converges. What relevant Theorem is used to conclude this? (d) What does the fact that {Sk} converges have to do with convergence of En=1 an? (e) Would part a) be true if an values could be any real numbers (so positive or negative)? If you say yes, then support your claim with a brief argument. If you say no, then give a counterexample, i.e. an example of a sequence {a.,} for which {S.} is not increasing.

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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trying to prove!). Suppose that 0 < an < bn for all n > 1, and define Sk = a1 + a2 + ... + ak. Suppose also that L =E1 bn (so the series converges to a sum L.) n=1 (a) Suppose that a, 2 0 for all n. How do we know that the sequence {Sk} is increasing? Show directly that Sk < Sk+1 (b) Explain why the sequence {Sk} is bounded. (c) Explain why {Sk} converges. What relevant Theorem is used to conclude this? (d) What does the fact that {Sk} converges have to do with convergence of En=1 an? (e) Would part a) be true if an values could be any real numbers (so positive or negative)? If you say yes, then support your claim with a brief argument. If you say no, then give a counterexample, i.e. an example of a sequence {am} for which {S½} is not increasing.
Transcribed Image Text:trying to prove!). Suppose that 0 < an < bn for all n > 1, and define Sk = a1 + a2 + ... + ak. Suppose also that L =E1 bn (so the series converges to a sum L.) n=1 (a) Suppose that a, 2 0 for all n. How do we know that the sequence {Sk} is increasing? Show directly that Sk < Sk+1 (b) Explain why the sequence {Sk} is bounded. (c) Explain why {Sk} converges. What relevant Theorem is used to conclude this? (d) What does the fact that {Sk} converges have to do with convergence of En=1 an? (e) Would part a) be true if an values could be any real numbers (so positive or negative)? If you say yes, then support your claim with a brief argument. If you say no, then give a counterexample, i.e. an example of a sequence {am} for which {S½} is not increasing.
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