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wu :: forums    riddles    medium (Moderators: Grimbal, william wu, towr, Eigenray, SMQ, Icarus, ThudnBlunder)    An Identity with binomial coefficients « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: An Identity with binomial coefficients  (Read 475 times) Aryabhatta Uberpuzzler     Gender: Posts: 1321 An Identity with binomial coefficients   « on: Jun 11th, 2008, 1:01pm » Quote Modify Not sure how easy/hard this is:   Show that if m is a positive integer, then   Sum (-1)m-r C(m,r)/(2r + 1)   Sum (-1)r C(m,r)/(2r + 1) = 2.4.6...(2m)/1.3.5....(2m+1)   where r ranges from 0 to m   and C(m,r) = m choose r.   (For instance if m = 2, the value is 8/15) « Last Edit: Jun 11th, 2008, 3:53pm by Aryabhatta » IP Logged Eigenray wu::riddles Moderator Uberpuzzler     Gender: Posts: 1948 Re: An Identity with binomial coefficients   « Reply #1 on: Jun 11th, 2008, 3:42pm » Quote Modify I think the sign is wrong when m is odd.  But both sides are (up to a sign) integral of (1-t2)m, t=0..1. IP Logged Aryabhatta Uberpuzzler     Gender: Posts: 1321 Re: An Identity with binomial coefficients   « Reply #2 on: Jun 11th, 2008, 3:52pm » Quote Modify Yes, you are right. I have edited the original post. Hopefully it is correct now.   And you gave the solution I had in mind. « Last Edit: Jun 11th, 2008, 3:54pm by Aryabhatta » IP Logged Aryabhatta Uberpuzzler     Gender: Posts: 1321 Re: An Identity with binomial coefficients   « Reply #3 on: Jun 11th, 2008, 4:21pm » Quote Modify Follow up question:   Find the series sum:   \Sum 1/C(2m+1,m)   where m ranges through integers from 0 to infinity.   C(2m+1,m) = 2m+1 choose m. IP Logged Eigenray wu::riddles Moderator Uberpuzzler     Gender: Posts: 1948 Re: An Identity with binomial coefficients   « Reply #4 on: Jun 11th, 2008, 9:45pm » Quote Modify That's a bit harder.  If we can solve the differential equation (4x-x2)y' = 2(x-1)y + 2, y(0)=1, we get y(1) = 2/3 + 43/27. IP Logged Aryabhatta Uberpuzzler     Gender: Posts: 1321 Re: An Identity with binomial coefficients   « Reply #5 on: Jun 12th, 2008, 9:18am » Quote Modify Or we could use the fact that   2.4.6....2m/1.3....(2m+1) = 22m/(m+1)C(2m+1,m)   IP Logged Eigenray wu::riddles Moderator Uberpuzzler     Gender: Posts: 1948 Re: An Identity with binomial coefficients   « Reply #6 on: Jun 12th, 2008, 9:59am » Quote Modify Ah, very nice: integrate 16/(3+t2)2.   In fact we can also say   1/C(2n+1,n) = 16/9* (-1/3)n(n+1)/(2n+1),   which is not obvious. « Last Edit: Jun 12th, 2008, 10:11am by Eigenray » IP Logged Aryabhatta Uberpuzzler     Gender: Posts: 1321 Re: An Identity with binomial coefficients   « Reply #7 on: Jun 12th, 2008, 10:37am » Quote Modify Yes!   Also, I guess you obtained that differential equation to solve for the generating function of 1/C(2n+1,n), which can also be given as an integral by using similar ideas.   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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