[math-fun] New (to me) proof of divergence of the harmonic series
Up until now, I only knew a couple of standard proofs that the harmonic series diverges: 1) 1 + 1/2 + (1/3 + 1/4) + (1/5 + ... + 1/8) + ... > 1/2 + 1/2 + 1/2 + 1/2 + ... 2) integral(1/x) = ln x I think that's about it ... But then I saw this very very cool proof idea in what was otherwise a mostly weak book of philosophy of mathematics ( http://www.amazon.com/How-Mathematicians-Think-Contradiction-Mathematics/dp/... if you are interested ) and I think it could be said something like this: Suppose it converges. Then let H = 1 + 1/2 + 1/3 + 1/4 + 1/5 + ... so 1/2 H = 1/2 + 1/4 + 1/6 + 1/8 + ... and, subtracting, H - 1/2 H = 1 + 1/3 + 1/5 + ... Finally comparing term-for-term, it's clear that 1/2 H > 1/2 H, contradiction, QED. Does anyone have an early citation for this proof, or know who might have thought it up? Byers doesn't give any citation for the idea in his book. Are there any major holes in the proof that need to be filled, or is this basically a good outline? (Since we assume convergence to get the contradiction, I think the manipulations in the middle are OK, right?) Thanks, --Joshua Zucker
Does anyone have an early citation for this proof, or know who might have thought it up? Byers doesn't give any citation for the idea in his book.
Dear Joshua, years ago I came up with the same proof and asked in a math forum pretty much the same questions you did. Nobody could name an early source but they confirmed that the proof works. Hopefully some other reader can shed some light on the question. (I conjecture that the proof has been thought of hundreds of times, finding an early citation is another thing) Another "classical" proof would be: If the sum converges, then, for some N, we have 1/N + 1/(N+1) + ... < 1/2. On the other hand, if we take the first N+1 terms of the series 1/N + 1/(N+1) + ... + 1/(2*N) > (N+1)/(2*N) > 1/2 (because we have (N+1) terms and all of them are bigger than 1/(2N)). Contradiction. Best wishes, Stefan
Googling I find: 31 proofs that the harmonic series converges are collected in these two papers: http://faculty.prairiestate.edu/skifowit/htdocs/harmapa.pdf http://faculty.prairiestate.edu/skifowit/htdocs/harm2.pdf Your proof is number 8 in the list. :-)
Those were some cute proofs! In a similar vein Robin Chapman gives 14 proofs of the more difficult fact that the sum of the reciprocals of the squares of the natural numbers equals pi squared over six. You can find them here: http://www.secamlocal.ex.ac.uk/people/staff/rjchapma/etc/zeta2.pdf -- On Fri, Mar 21, 2008 at 6:56 AM, Edwin Clark <eclark@math.usf.edu> wrote:
Googling I find:
31 proofs that the harmonic series converges are collected in these two papers:
http://faculty.prairiestate.edu/skifowit/htdocs/harmapa.pdf
http://faculty.prairiestate.edu/skifowit/htdocs/harm2.pdf
Your proof is number 8 in the list. :-)
Hello Math-Fun, Find the first eleven consecutive integers whose digit sum is a substring of themselves. http://www.research.att.com/~njas/sequences/A052018 <http://www.research.att.com/~njas/sequences/A052018> Best, E.
You need absolute convergence for the rearrangements implied by the expression H - H/2 to be valid. Since we're discussing a series of all positive terms, and assuming it converges, this is OK. However, the analysis required to prove that "absolute convergence makes the rearrangements OK" dwarfs all three of the proofs. Rich ________________________________________ From: math-fun-bounces@mailman.xmission.com [math-fun-bounces@mailman.xmission.com] On Behalf Of Joshua Zucker [joshua.zucker@gmail.com] Sent: Thursday, March 20, 2008 6:18 PM To: math-fun Subject: [math-fun] New (to me) proof of divergence of the harmonic series Up until now, I only knew a couple of standard proofs that the harmonic series diverges: 1) 1 + 1/2 + (1/3 + 1/4) + (1/5 + ... + 1/8) + ... > 1/2 + 1/2 + 1/2 + 1/2 + ... 2) integral(1/x) = ln x I think that's about it ... But then I saw this very very cool proof idea in what was otherwise a mostly weak book of philosophy of mathematics ( http://www.amazon.com/How-Mathematicians-Think-Contradiction-Mathematics/dp/... if you are interested ) and I think it could be said something like this: Suppose it converges. Then let H = 1 + 1/2 + 1/3 + 1/4 + 1/5 + ... so 1/2 H = 1/2 + 1/4 + 1/6 + 1/8 + ... and, subtracting, H - 1/2 H = 1 + 1/3 + 1/5 + ... Finally comparing term-for-term, it's clear that 1/2 H > 1/2 H, contradiction, QED. Does anyone have an early citation for this proof, or know who might have thought it up? Byers doesn't give any citation for the idea in his book. Are there any major holes in the proof that need to be filled, or is this basically a good outline? (Since we assume convergence to get the contradiction, I think the manipulations in the middle are OK, right?) Thanks, --Joshua Zucker _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (6)
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Edwin Clark -
Eric Angelini -
James Buddenhagen -
Joshua Zucker -
Schroeppel, Richard -
Stefan Steinerberger