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You are not logged in. #1 20070917 00:27:03
Prime seriesI was thinking about the infinite series that goes 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ..., where the nth term is one divided by the nth prime. Why did the vector cross the road? It wanted to be normal. #2 20070917 01:28:01
Re: Prime seriesBased on what you said about f(k+1)  f(k), wouldn't it converge since the distance between between primes increases as k increases? But that's just my intuition speaking there, perhaps primes don't thin out sufficiently at infinity. I'd be interested in what it converges to though... #3 20070917 02:06:35
Re: Prime seriesI think I read on wikipeida that it diverges, not that they're a respectable source... Last edited by bossk171 (20070917 02:09:23) There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction. #4 20070917 02:57:42
Re: Prime serieshttp://www.maa.org/editorial/euler/How% … primes.pdf Last edited by bossk171 (20070917 02:58:27) There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction. #5 20070917 03:29:01
Re: Prime seriesAnd silly me was trying to estimate its convergence. Until I read bossk171's post I thought it would converge to e. #6 20070917 04:50:28
Re: Prime seriesThis proof is a bit easier to understand. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #7 20070917 05:16:14
Re: Prime seriesI only sorta get that. Last edited by bossk171 (20070917 05:20:32) There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction. #8 20070917 05:29:31
Re: Prime series
Which statement do you not get? "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #9 20070917 06:21:38
Re: Prime series"Well, there's an elementary theorem of calculus that a product (1a1)...(1ak)... with ak>0 converges to a nonzero value iff the sum a1+...+ak+... converges" There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction. #10 20070917 13:05:46
Re: Prime seriesAh, then you do follow this proof, it's just a proof for theorem used in it which you don't. I too, have yet to see proof of that theorem. Anyone care to try or dig one up? "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #11 20070917 13:47:19
Re: Prime series
I'm not sure if I've hear the phrase "open the brackets" before. Does this mean something common and it's just the slang I'm stuck on, or is this something I haven't learned yet? There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction. #12 20070917 15:14:02
Re: Prime series"Open the brackets" simply means reorder the terms. So long as you are summing all positive terms, any permutation (reordering) will work, so long as all terms eventually appear in the new sequence. But when you come to alternating series, the same isn't true. This is why you must be careful. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #13 20070918 01:12:30
Re: Prime seriesYeah, thanks! it all makes sense now. Except for that initial theorm. Last edited by bossk171 (20070918 01:12:46) There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction. 