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3. Prove the following: Theorem 2 (Ratio Test). Let an be a series with nonzero terms. Then Σan converges absolutely if lim sup • Σ an diverges if lim inf an+1 an >1 +1₁ an +1 an < 1

3. Prove the following: Theorem 2 (Ratio Test). Let an be a series with nonzero terms. Then Σan converges absolutely if lim sup • Σ an diverges if lim inf an+1 an >1 +1₁ an +1 an < 1

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 64E
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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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