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.
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.
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: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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