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wu :: forums    riddles    putnam exam (pure math) (Moderators: Grimbal, Icarus, Eigenray, towr, SMQ, william wu)    f(x) = 1/(1+e^x) « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: f(x) = 1/(1+e^x)  (Read 650 times) william wu wu::riddles Administrator     Gender: Posts: 1291 f(x) = 1/(1+e^x)   « on: Oct 24th, 2004, 12:06am » Quote Modify Given f(x) = 1/(1+ex), find f(100)(0).     [ The f(n) notation denotes repeated application of the function. So f(2)(x) = f(f(x)). ]     Edit: corrected a typo « Last Edit: Oct 24th, 2004, 10:24am by william wu » IP Logged [ wu ] : http://wuriddles.com / http://forums.wuriddles.com Barukh Uberpuzzler     Gender: Posts: 2276 Re: f(x) = 1/(1+e^x)   « Reply #1 on: Oct 24th, 2004, 12:49am » Quote Modify Isn't it just the solution of the equation x = 1/(1+e^x)?     IP Logged william wu wu::riddles Administrator     Gender: Posts: 1291 Re: f(x) = 1/(1+e^x)   « Reply #2 on: Oct 24th, 2004, 10:26am » Quote Modify Sorry, I had a typo: it should read f(100)(0).   I should warn beforehand that I haven't been able to solve this myself; hopefully the problem is legit. « Last Edit: Oct 24th, 2004, 10:26am by william wu » IP Logged [ wu ] : http://wuriddles.com / http://forums.wuriddles.com Grimbal wu::riddles Moderator Uberpuzzler     Gender: Posts: 7527 Re: f(x) = 1/(1+e^x)   « Reply #3 on: Oct 24th, 2004, 3:26pm » Quote Modify It still tends quite fast to the solution of x=1/(1+e^x).  It is not there yet, but close enough. IP Logged Icarus wu::riddles Moderator Uberpuzzler Boldly going where even angels fear to tread.     Gender: Posts: 4863 Re: f(x) = 1/(1+e^x)   « Reply #4 on: Oct 24th, 2004, 3:26pm » Quote Modify According to Excel, it is 0.401058138.... Repeated application converges to this value fairly fast. As Barukh indicates, this is the solution to x = 1/(1+ex). I would imagine it is highly unlikely that f(100)(0) is exactly the value of x solving the equation. It is very close, however.   Whether or not there is "nice" expression for the number is another matter. IP Logged "Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed? " - Anonymous 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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