Riddle Me This, Batman!
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Electric Sheep One
Member # 3431
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written Monday, August 21 2006 09:10
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And I guess I'm asking, "Are you sure?" The answer you've given is old hat; I'm asking for a rethinking, not just a reiteration. If 0.999... is not by definition an infinite series, what is its definition? And if it is, isn't it indeed the limit as 1 is approached from below? -------------------- We're not doing cool. We're doing pretty. Posts: 3335 | Registered: Thursday, September 4 2003 07:00 |
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Post Navel Trauma ^_^
Member # 67
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written Monday, August 21 2006 09:52
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Inventing notation here... 0.999 is indeed lim{N->inf} sum(1,N, 9/(10^n)) 1/(1-sum(0,N, 9/(10^n)) does tend to positive infinity as N tends to infinity. However, 1/(1-lim{N->inf}sum(1,N,9/(10^n))) is undefined, because lim{N->inf}sum(1,N,9/(10^n)) = 1, and 1/0 is not defined. It's often possible to be sloppy when dealing with limits, and say things like "1/lim(thing) = lim(1/thing)", but you have be more careful around points where some of your functions aren't defined, or are discontinuous, or whatever. This is one of the times when you can't be sloppy. -------------------- Barcoorah: I even did it to a big dorset ram. desperance.net - Don't follow this link Posts: 1798 | Registered: Thursday, October 4 2001 07:00 |
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Electric Sheep One
Member # 3431
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written Monday, August 21 2006 13:05
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Ah; I see your point. Thanks for the very clear answer. I guess I'm too used to physics, where one indeed often has to be careful about orders of limits, but where the right thing to do is almost always to keep everything finite until the final answer, and only then take the limit. Our limit ordering problems almost always arise because huge but finite quantities have been approximated as infinite, and although you can sometimes retain the infinite limits and come out okay with fancy footwork, the really correct thing to do is to remember that in fact it's all finite. [ Tuesday, August 22, 2006 02:39: Message edited by: Student of Trinity ] -------------------- We're not doing cool. We're doing pretty. Posts: 3335 | Registered: Thursday, September 4 2003 07:00 |
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Agent
Member # 2759
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written Tuesday, August 22 2006 10:24
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While we're on the subject of mathematics, did you hear the one about the guy who refused a Fields Medal? Grigory Perelman declines top prize Chance'd be a fine thing... -------------------- "I can't read this thread with that image. But then, that's not a complaint." -Scorpius Geneforge 4 stuff. Also, everything I know about Avernum | Avernum 2 | Avernum 3 | Avernum 4 Posts: 1104 | Registered: Monday, March 10 2003 08:00 |