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Theorem 1 (Limit Comparison Test). Let an and bn be sequences such that an ≥ 0 and bn > 0 for all n € N. If lim (%) = = c with c a positive real number, then either Ean and Ebn both converge or both diverge.

Theorem 1 (Limit Comparison Test). Let an and bn be sequences such that an ≥ 0 and bn > 0 for all n € N. If lim (%) = = c with c a positive real number, then either Ean and Ebn both converge or both diverge.

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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1. We have shown that if a sequence is unbounded and increasing, then it diverges to infinity. The same is true if we weaken the hypothesis to be eventually increasing¹. Prove or disprove the converse: If lim(sn) = ∞, then (sn) is unbounded and eventually increasing. 2. Prove the following: Theorem 1 (Limit Comparison Test). Let an and Ebn be sequences such an that an 20 and bn > 0 for all n € N. If lim =) = = c with c a positive real number, then either Σan and bn both converge or both diverge. 3. Prove the following: Theorem 2 (Ratio Test). Let an be a series with nonzero terms. Then • Σan converges absolutely if lim sup • Σan diverges if lim infan¹|>1 an+1 an <1
Transcribed Image Text:1. We have shown that if a sequence is unbounded and increasing, then it diverges to infinity. The same is true if we weaken the hypothesis to be eventually increasing¹. Prove or disprove the converse: If lim(sn) = ∞, then (sn) is unbounded and eventually increasing. 2. Prove the following: Theorem 1 (Limit Comparison Test). Let an and Ebn be sequences such an that an 20 and bn > 0 for all n € N. If lim =) = = c with c a positive real number, then either Σan and bn both converge or both diverge. 3. Prove the following: Theorem 2 (Ratio Test). Let an be a series with nonzero terms. Then • Σan converges absolutely if lim sup • Σan diverges if lim infan¹|>1 an+1 an <1
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