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 00 converges. n=1
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 00 converges. n=1
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: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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