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wu :: forums    riddles    putnam exam (pure math) (Moderators: SMQ, Eigenray, Grimbal, Icarus, william wu, towr)    Must this series converge? « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: Must this series converge?  (Read 506 times) ecoist Senior Riddler     Gender: Posts: 405 Must this series converge?   « on: Apr 15th, 2006, 8:58am » Quote Modify Let {an} and {bn} be monotone increasing unbounded sequences of positive real numbers.  Must the series   [sum]an-bn   always converge? IP Logged Barukh Uberpuzzler     Gender: Posts: 2276 Re: Must this series converge?   « Reply #1 on: Apr 20th, 2006, 9:15am » Quote Modify If I understand correctly, the answer is no: take bn = log log(n), and anbn = n. For an appropriate base, both sequences are strictly increasing. IP Logged ecoist Senior Riddler     Gender: Posts: 405 Re: Must this series converge?   « Reply #2 on: Apr 20th, 2006, 2:22pm » Quote Modify Here's a different, more specific solution.  Let bn=sqr(ln n) and an=ebn, for n>1.  Then an-bn=1/n, and the series is the divergent harmonic series (as is Barukh's solution). IP Logged Pages: 1  Reply Notify of replies Send Topic Print Forum Jump: ----------------------------- riddles -----------------------------  - easy   - medium   - hard   - what am i   - what happened   - microsoft   - cs => putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth   - complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »

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