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diverges. n=1

diverges. n=1

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section: Chapter Questions
Problem 63RE
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How do you solve the third bullet point?

Let an > 0 and assume that > an diverges. Write Sn = a1 + a2 + • ·+ an . Prove the following statements. n=1 an diverges. 1+an n=1 converges. an diverges. п-1 Hint1: an+1/Sn+1+…+am/Sm > (Sm – Sn)/Sm (justify it!). Hint2: if lim Sn-1/Sn = 0, it is obvious; otherwise there is some c > 0 such that Sn–1/Sn > c, then use a definite integral of 1/x to bound (Sn – Sn-1)/Sn-1.
Transcribed Image Text:Let an > 0 and assume that > an diverges. Write Sn = a1 + a2 + • ·+ an . Prove the following statements. n=1 an diverges. 1+an n=1 converges. an diverges. п-1 Hint1: an+1/Sn+1+…+am/Sm > (Sm – Sn)/Sm (justify it!). Hint2: if lim Sn-1/Sn = 0, it is obvious; otherwise there is some c > 0 such that Sn–1/Sn > c, then use a definite integral of 1/x to bound (Sn – Sn-1)/Sn-1.
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