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Topic: harmonic (Read 4623 times) |
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srn437
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What is 1-1/2 +1/3 -1/4...
what about 1/9 +1/19+1/29...(all that have a nine anywhere in the denominator)
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Sameer
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Re: harmonic
« Reply #1 on: Oct 1st, 2007, 8:43pm » |
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First is  (1) 
Second is complemented depleted/Kempner series.
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| « Last Edit: Oct 1st, 2007, 8:52pm by Sameer » |
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"Obvious" is the most dangerous word in mathematics.
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srn437
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Re: harmonic
« Reply #2 on: Oct 1st, 2007, 9:06pm » |
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I meant what are their numerical values.
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Sameer
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Re: harmonic
« Reply #3 on: Oct 1st, 2007, 9:19pm » |
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on Oct 1st, 2007, 9:06pm, srn347 wrote:| I meant what are their numerical values. |
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First one is another way of saying what is its numerical value!! That's the way how we do it!!
For second I would say 80 http://www.qbyte.org/puzzles/p072s.html
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| « Last Edit: Oct 1st, 2007, 9:48pm by Sameer » |
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"Obvious" is the most dangerous word in mathematics.
--Bell, Eric Temple
Proof is an idol before which the mathematician tortures himself.
Sir Arthur Eddington, quoted in Bridges to Infinity
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srn437
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Re: harmonic
« Reply #4 on: Oct 1st, 2007, 9:24pm » |
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The n thing doesn't tell me the number. the 80 was on the right track, but out of the loop(the track shouldn't be shaped like a loop anyway). Go back to line 10(in binary).
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Sameer
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Re: harmonic
« Reply #5 on: Oct 1st, 2007, 9:42pm » |
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on Oct 1st, 2007, 9:24pm, srn347 wrote:| The n thing doesn't tell me the number |
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It does if you know Dirichlet!!
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"Obvious" is the most dangerous word in mathematics.
--Bell, Eric Temple
Proof is an idol before which the mathematician tortures himself.
Sir Arthur Eddington, quoted in Bridges to Infinity
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srn437
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the dark lord rises again....
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Re: harmonic
« Reply #6 on: Oct 2nd, 2007, 6:28am » |
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was that supposed to be the remnien zeta? regardless, I require an actual answer.
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pex
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Re: harmonic
« Reply #7 on: Oct 2nd, 2007, 6:50am » |
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on Oct 2nd, 2007, 6:28am, srn347 wrote:| was that supposed to be the remnien zeta? regardless, I require an actual answer. |
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Not many people write eta when they mean zeta. Sameer's answer is correct: the Dirichlet Eta function evaluated at 1.
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srn437
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Re: harmonic
« Reply #8 on: Oct 2nd, 2007, 7:21am » |
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Nope. I don't know how you got an answer of 1...
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pex
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Re: harmonic
« Reply #9 on: Oct 2nd, 2007, 7:26am » |
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on Oct 2nd, 2007, 7:21am, srn347 wrote:| Nope. I don't know how you got an answer of 1... |
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He doesn't, he gets eta(1).
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Grimbal
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Re: harmonic
« Reply #10 on: Oct 2nd, 2007, 7:27am » |
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The Dirichlet Eta function evaluated at 1 evaluates as ln(2).
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| « Last Edit: Oct 2nd, 2007, 7:27am by Grimbal » |
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srn437
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Re: harmonic
« Reply #11 on: Oct 2nd, 2007, 7:27am » |
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Does that mean zeta 1? That is incorrect.
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pex
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Re: harmonic
« Reply #12 on: Oct 2nd, 2007, 7:28am » |
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on Oct 2nd, 2007, 7:27am, srn347 wrote:| Does that mean zeta 1? That is incorrect. |
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No, it means exactly what it says.
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ThudnBlunder
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Re: harmonic
« Reply #13 on: Oct 2nd, 2007, 10:06am » |
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347, your series 1 - 1/2 + 1/3 - 1/4 +..... = loge2 = 0.63452380952380952380952380952381...
Edit: loge2 = 0.69314718055994530941723212145818...
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| « Last Edit: Oct 3rd, 2007, 9:40am by ThudnBlunder » |
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THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
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srn437
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Re: harmonic
« Reply #14 on: Oct 2nd, 2007, 4:32pm » |
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Nope. there is fallacy in your statement.
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pex
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Re: harmonic
« Reply #15 on: Oct 2nd, 2007, 4:40pm » |
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on Oct 2nd, 2007, 4:32pm, srn347 wrote:| Nope. there is fallacy in your statement. |
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Until "loge2", ThudanBlunder is correct (although you may be more familiar with seeing it written as ln 2), but I don't know where he got his decimal expression. Maple tells me it's 0.693147180559945309417232121458176568...
By the way, also eta(1) = ln 2.
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| « Last Edit: Oct 2nd, 2007, 4:52pm by pex » |
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pex
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Re: harmonic
« Reply #16 on: Oct 2nd, 2007, 5:02pm » |
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Just for completeness, the second series diverges. Proof:
Its "complement", the series summing the reciprocals of all integers containing no nines, is
(1/1 + 1/2 + ... + 1/8)
+ (1/10 + 1/11 + ... + 1/18 + 1/20 + ... + 1/88)
+ (1/100 + ... + 1/888)
+ ...
The n-th bracketed term contains 8*9n-1 numbers which are reciprocals of n-digit integers. Thus, the sum is less than
8*(1/1) + 8*9*(1/10) + 8*92*(1/100) + ...
= 8*sum{n=0 to infinity} (9/10)n
= 8 * 1/(1-9/10) = 80,
which means that it converges.
We know that the sum of reciprocals of all integers (1/1 + 1/2 + 1/3 + ...) diverges, and we know that the sum of the reciprocals of all integers that contain no nines converges. This means that the part that is left over, the sum of the reciprocals of all integers that do contain nines, must diverge.
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| « Last Edit: Oct 2nd, 2007, 5:03pm by pex » |
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srn437
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Re: harmonic
« Reply #17 on: Oct 2nd, 2007, 8:46pm » |
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It is not log or ln of whatever. should I tell you what it is?
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Obob
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Re: harmonic
« Reply #18 on: Oct 2nd, 2007, 10:19pm » |
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Its very arrogant of you to tell all these people who are much smarter than you that they are wrong. Especially since they are in fact correct, and you don't even know the math involved.
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srn437
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Re: harmonic
« Reply #19 on: Oct 2nd, 2007, 11:38pm » |
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The first one is zero, because it is the harmonic series(infinity) minus itself. 1-1/2+1/3...=1-(1-1/2)+1/3-(1/2-1/4)...=(1-1)+(1/2-1/2)+(1/3)-1/3... If you think it's arrogant, try taking ritalin. if you are offended, blame the one whom the phase originated from.
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pex
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Re: harmonic
« Reply #20 on: Oct 3rd, 2007, 12:40am » |
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on Oct 2nd, 2007, 11:38pm, srn347 wrote:| The first one is zero, because it is the harmonic series(infinity) minus itself. 1-1/2+1/3...=1-(1-1/2)+1/3-(1/2-1/4)...=(1-1)+(1/2-1/2)+(1/3)-1/3... If you think it's arrogant, try taking ritalin. if you are offended, blame the one whom the phase originated from. |
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Wow. This could just be one of the most stupid things I've ever seen.
One. "Infinity minus infinity" is not well-defined; by no means does it have to be zero.
Two. You are trying to use the commutative property (look it up) of addition in a conditionally convergent series (look that one up too). This generally does not work.
Three. Any decent textbook or website that treats limits will tell you that the value is ln 2.
Four. So did multiple math-inclined (to say the least) people above.
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towr
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Re: harmonic
« Reply #21 on: Oct 3rd, 2007, 2:25am » |
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Five. Try the start of the sequence on your trusty calculator, and you'll see that in no way does it tend to 0.
Six. Ritalin does not work against other people's arrogance, so no matter how much ritalin any of us were to take, you'd still be an arrogant little bugger.
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| « Last Edit: Oct 3rd, 2007, 2:26am by towr » |
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Wikipedia, Google, Mathworld, Integer sequence DB
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srn437
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Re: harmonic
« Reply #22 on: Oct 3rd, 2007, 6:55am » |
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Try solving the second series.
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pex
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Re: harmonic
« Reply #23 on: Oct 3rd, 2007, 7:20am » |
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on Oct 3rd, 2007, 6:55am, srn347 wrote:| Try solving the second series. |
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Look further up in the thread - it diverges. But you're not going to believe this anyway.
Edit: perhaps it's nice if I mention that "my" proof of this fact was inspired rather heavily by NickH's solution in the link Sameer posted. Night-time plagiarism...
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| « Last Edit: Oct 3rd, 2007, 7:25am by pex » |
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Sameer
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Re: harmonic
« Reply #24 on: Oct 3rd, 2007, 7:58am » |
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*sigh* how come you didn't tell us you didn't want the solution that everyone knows but the one that you concocted up... it would have made our thinking so much simpler than these god knows what eta, zeta, etc..
Edit: I tried giving benefit of doubt and encouraging good posts, but it seems it is pointless.
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| « Last Edit: Oct 3rd, 2007, 8:04am by Sameer » |
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"Obvious" is the most dangerous word in mathematics.
--Bell, Eric Temple
Proof is an idol before which the mathematician tortures himself.
Sir Arthur Eddington, quoted in Bridges to Infinity
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