[math-fun] Bogus proofs using rearrangement of conditionally convergent series
Can anyone provide a favorite web-reference? For my upcoming Mathematical Enchantments column, I composed a proof that 1 - 1/2 + 1/3 - 1/4 + ... is both positive and zero, but it really doesn't fit into the (already over-long) article, so I'd like to outsource the job of showing "You can get nonsensical results by rearranging conditionally convergent series" to some website that already exists. Jim Propp
="James Propp" <jamespropp@gmail.com>
Can anyone provide a favorite web-reference?
For my upcoming Mathematical Enchantments column, I composed a proof that 1 - 1/2 + 1/3 - 1/4 + ... is both positive and zero, but it really doesn't fit into the (already over-long) article, so I'd like to outsource the job of showing "You can get nonsensical results by rearranging conditionally convergent series" to some website that already exists.
Best wishes on your citation hunt--I'll look forward to your article! However you might want to consider the risk of prejudicing your less-sophisticated readers with pejoratives like "nonsensical". For instance 1+2+4+8+16+32+... is arguably both plus infinity and minus one. Virtually all modern computing device architectures rely on such "nonsense". There's fragile nuance in what exactly we mean by "...".
This is trivial to do, so you don't need a web-reference. But please don't label a justification for 1 - 1/2 + 1/3 - 1/4 + ... = 0. a "proof". People have already deeply bewildered by the recent kerfuffle about whether 1 + 1/2 + 1/3 + 1/4 + ... = -1/12 (following a YouTube video purporting to prove this) to the point that the Wikipedia article on this will not budge on the point that this is merely "true", and Terry Tao remained steadfastly ambiguous on the subject. —Dan
On Sep 9, 2015, at 10:03 AM, James Propp <jamespropp@gmail.com> wrote:
Can anyone provide a favorite web-reference?
For my upcoming Mathematical Enchantments column, I composed a proof that 1 - 1/2 + 1/3 - 1/4 + ... is both positive and zero, but it really doesn't fit into the (already over-long) article, so I'd like to outsource the job of showing "You can get nonsensical results by rearranging conditionally convergent series" to some website that already exists.
Dan, can you give a reference to which Wikipedia article has the solecism you mention? I can't see a trace of such a claim in the article "Harmonic Series (mathematics)", which was my first guess. On Wed, Sep 9, 2015 at 2:27 PM, Dan Asimov <dasimov@earthlink.net> wrote:
This is trivial to do, so you don't need a web-reference.
But please don't label a justification for
1 - 1/2 + 1/3 - 1/4 + ... = 0.
a "proof".
People have already deeply bewildered by the recent kerfuffle about whether
1 + 1/2 + 1/3 + 1/4 + ... = -1/12
(following a YouTube video purporting to prove this) to the point that the Wikipedia article on this will not budge on the point that this is merely "true", and Terry Tao remained steadfastly ambiguous on the subject.
—Dan
On Sep 9, 2015, at 10:03 AM, James Propp <jamespropp@gmail.com> wrote:
Can anyone provide a favorite web-reference?
For my upcoming Mathematical Enchantments column, I composed a proof that 1 - 1/2 + 1/3 - 1/4 + ... is both positive and zero, but it really doesn't fit into the (already over-long) article, so I'd like to outsource the job of showing "You can get nonsensical results by rearranging conditionally convergent series" to some website that already exists.
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Actually, I meant the series 1 + 2 + 3 + ... rather than the harmonic series. Sorry about that. This is the Wikipedia article https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_⋯ <https://en.wikipedia.org/wiki/1_+_2_+_3_+_4_+_%E2%8B%AF> Plus, I was reporting from memory — which was correct not too long ago — but apparently the non-Yahoos have finally won and the article no longer insists that a false statement is true. —Dan
On Sep 9, 2015, at 11:33 AM, Allan Wechsler <acwacw@gmail.com> wrote:
Dan, can you give a reference to which Wikipedia article has the solecism you mention? I can't see a trace of such a claim in the article "Harmonic Series (mathematics)", which was my first guess.
On Wed, Sep 9, 2015 at 2:27 PM, Dan Asimov <dasimov@earthlink.net> wrote:
This is trivial to do, so you don't need a web-reference.
But please don't label a justification for
1 - 1/2 + 1/3 - 1/4 + ... = 0.
a "proof".
People have already deeply bewildered by the recent kerfuffle about whether
1 + 1/2 + 1/3 + 1/4 + ... = -1/12
(following a YouTube video purporting to prove this) to the point that the Wikipedia article on this will not budge on the point that this is merely "true", and Terry Tao remained steadfastly ambiguous on the subject.
—Dan
On Sep 9, 2015, at 10:03 AM, James Propp <jamespropp@gmail.com> wrote:
Can anyone provide a favorite web-reference?
For my upcoming Mathematical Enchantments column, I composed a proof that 1 - 1/2 + 1/3 - 1/4 + ... is both positive and zero, but it really doesn't fit into the (already over-long) article, so I'd like to outsource the job of showing "You can get nonsensical results by rearranging conditionally convergent series" to some website that already exists.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
You could just use Mathematica: For consecutive n, these three terms enumerate -(-)^n/n, Out[567]= {1/(-3 + 4 n), 1/(-1 + 4 n), -1/(2 n)} but taking two odds for every even: In[568]:= Table[%, {n, 5}] Out[568]= {{1, 1/3, -1/2}, {1/5, 1/7, -1/4}, {1/9, 1/11, -1/6}, {1/13, 1/15, -1/8}, {1/17, 1/19, -1/10}} Adding them all up, In[569]:= Inactive[Sum][Total[%%], {n, \[Infinity]}] Out[569]= Inactive[Sum][-1/(2*n) + 1/(-3 + 4*n) + 1/(-1 + 4*n), {n, Infinity}] In[570]:= Activate[%] Out[570]= Log[8]/2 I.e., 3/2 of the unpermuted series. --rwg On 2015-09-09 10:03, James Propp wrote:
Can anyone provide a favorite web-reference?
For my upcoming Mathematical Enchantments column, I composed a proof that 1 - 1/2 + 1/3 - 1/4 + ... is both positive and zero, but it really doesn't fit into the (already over-long) article, so I'd like to outsource the job of showing "You can get nonsensical results by rearranging conditionally convergent series" to some website that already exists.
Jim Propp
Mike Hirschhorn reminds me that there are infinitely many rearrangements of 1 - 1/2 + 1/3 - 1/4 + ... that sum to 42. --rwg On 2015-09-09 19:19, rwg wrote:
You could just use Mathematica: For consecutive n, these three terms enumerate -(-)^n/n,
Out[567]= {1/(-3 + 4 n), 1/(-1 + 4 n), -1/(2 n)}
but taking two odds for every even:
In[568]:= Table[%, {n, 5}]
Out[568]= {{1, 1/3, -1/2}, {1/5, 1/7, -1/4}, {1/9, 1/11, -1/6}, {1/13, 1/15, -1/8}, {1/17, 1/19, -1/10}}
Adding them all up, In[569]:= Inactive[Sum][Total[%%], {n, \[Infinity]}]
Out[569]= Inactive[Sum][-1/(2*n) + 1/(-3 + 4*n) + 1/(-1 + 4*n), {n, Infinity}]
In[570]:= Activate[%]
Out[570]= Log[8]/2
I.e., 3/2 of the unpermuted series. --rwg
On 2015-09-09 10:03, James Propp wrote:
Can anyone provide a favorite web-reference?
For my upcoming Mathematical Enchantments column, I composed a proof that 1 - 1/2 + 1/3 - 1/4 + ... is both positive and zero, but it really doesn't fit into the (already over-long) article, so I'd like to outsource the job of showing "You can get nonsensical results by rearranging conditionally convergent series" to some website that already exists.
Jim Propp
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
By suitable rearrangement, a conditionally convergent series can sum to any arbitrarily chosen value. Furthermore, a rearrangement exists such that the successive sums have any arbitrarily chosen limsup and liminf. -- Gene From: rwg <rwg@sdf.org> To: math-fun <math-fun@mailman.xmission.com> Sent: Wednesday, September 9, 2015 9:50 PM Subject: Re: [math-fun] Bogus proofs using rearrangement of conditionally convergent series Mike Hirschhorn reminds me that there are infinitely many rearrangements of 1 - 1/2 + 1/3 - 1/4 + ... that sum to 42. --rwg On 2015-09-09 19:19, rwg wrote:
You could just use Mathematica: For consecutive n, these three terms enumerate -(-)^n/n,
Out[567]= {1/(-3 + 4 n), 1/(-1 + 4 n), -1/(2 n)}
but taking two odds for every even:
In[568]:= Table[%, {n, 5}]
Out[568]= {{1, 1/3, -1/2}, {1/5, 1/7, -1/4}, {1/9, 1/11, -1/6}, {1/13, 1/15, -1/8}, {1/17, 1/19, -1/10}}
Adding them all up, In[569]:= Inactive[Sum][Total[%%], {n, \[Infinity]}]
Out[569]= Inactive[Sum][-1/(2*n) + 1/(-3 + 4*n) + 1/(-1 + 4*n), {n, Infinity}]
In[570]:= Activate[%]
Out[570]= Log[8]/2
I.e., 3/2 of the unpermuted series. --rwg
On 2015-09-09 10:03, James Propp wrote:
Can anyone provide a favorite web-reference?
For my upcoming Mathematical Enchantments column, I composed a proof that 1 - 1/2 + 1/3 - 1/4 + ... is both positive and zero, but it really doesn't fit into the (already over-long) article, so I'd like to outsource the job of showing "You can get nonsensical results by rearranging conditionally convergent series" to some website that already exists.
Jim Propp
That was by far the most interesting thing I learned in 18.03. —Dan
On Sep 10, 2015, at 9:23 AM, Eugene Salamin via math-fun <math-fun@mailman.xmission.com> wrote:
By suitable rearrangement, a conditionally convergent series can sum to any arbitrarily chosen value. Furthermore, a rearrangement exists such that the successive sums have any arbitrarily chosen limsup and liminf.
On 2015-09-10 10:44, Dan Asimov wrote:
That was by far the most interesting thing I learned in 18.03.
—Dan
On Sep 10, 2015, at 9:23 AM, Eugene Salamin via math-fun <math-fun@mailman.xmission.com> wrote:
By suitable rearrangement, a conditionally convergent series can sum to any arbitrarily chosen value. Furthermore, a rearrangement exists such that the successive sums have any arbitrarily chosen limsup and liminf.
In fact, when limsup < liminf, it's *extremely* interesting. --rwg
For fixed p in (0,1], consider the formal series S(p) = Sum_{1 <= n < oo} e_n / n^p where each e_n = +-1 independently at random. Let f(p) := Prob(S(p) converges) Puzzle: Find f(p). —Dan
On Sep 10, 2015, at 11:13 AM, rwg <rwg@sdf.org> wrote:
On 2015-09-10 10:44, Dan Asimov wrote:
That was by far the most interesting thing I learned in 18.03. —Dan
On Sep 10, 2015, at 9:23 AM, Eugene Salamin via math-fun <math-fun@mailman.xmission.com> wrote: By suitable rearrangement, a conditionally convergent series can sum to any arbitrarily chosen value. Furthermore, a rearrangement exists such that the successive sums have any arbitrarily chosen limsup and liminf.
In fact, when limsup < liminf, it's *extremely* interesting. --rwg
Just rearrange that the set of terms of the series be the empty set.
Aha. Use a conditionally convergent, telescoping series: Instead of approaching the infinite sum via In[600]:= Sum[a[k], {k, 7}] Out[600]= a[1] + a[2] + a[3] + a[4] + a[5] + a[6] + a[7] In[601]:= Sum[a[k], {k, 9}] Out[601]= a[1] + a[2] + a[3] + a[4] + a[5] + a[6] + a[7] + a[8] + a[9] etc., take two odd terms for every even: In[596]:= Sum[a[2 k] + a[4 k - 3] + a[4 k - 1], {k, 3}] Out[596]= a[1] + a[2] + a[3] + a[4] + a[5] + a[6] + a[7] + a[9] + a[11] In[597]:= Sum[a[2 k] + a[4 k - 3] + a[4 k - 1], {k, 4}] Out[597]= a[1] + a[2] + a[3] + a[4] + a[5] + a[6] + a[7] + a[8] + a[9] + a[11] + a[13] + a[15] which, sooner or later, counts each term exactly once. Now put a(k) := (-1)^k/k - (-1)^(k+1)/(k+1) so that a[1], a[2], a[3], a[4], ... becomes -1-1^2/2, 1^2/2+1^3/3, -1^3/3-1^4/4, 1^4/4+1^5/5, . . . n terms of which clearly telescope to -1 + (-1)^n/(n+1), which clearly -> -1 as n-> oo. But the alternate, three-at-a-time summand simplifies to a[2 k] + a[-3 + 4 k] + a[-1 + 4 k] = 1/(1 - 4 k) + 1/(2 - 4 k) + 1/(3 - 4 k) + 1/(4 k) + 1/(1 + 2 k) This sum does *not* telescope, and according to Mathematica, converges to -1 - Log[2] ~ -1.69314718 < -1 (QED. This whole construction might be clearer with a[k] -> -a[k] := (-1)^(k+1)/(k+1) - (-1)^k/k .) --rwg On 2015-09-10 09:23, Eugene Salamin via math-fun wrote:
By suitable rearrangement, a conditionally convergent series can sum to any arbitrarily chosen value. Furthermore, a rearrangement exists such that the successive sums have any arbitrarily chosen limsup and liminf.
-- Gene
From: rwg <rwg@sdf.org> To: math-fun <math-fun@mailman.xmission.com> Sent: Wednesday, September 9, 2015 9:50 PM Subject: Re: [math-fun] Bogus proofs using rearrangement of conditionally convergent series
Mike Hirschhorn reminds me that there are infinitely many rearrangements of 1 - 1/2 + 1/3 - 1/4 + ... that sum to 42. --rwg
On 2015-09-09 19:19, rwg wrote:
You could just use Mathematica: For consecutive n, these three terms enumerate -(-)^n/n,
Out[567]= {1/(-3 + 4 n), 1/(-1 + 4 n), -1/(2 n)}
but taking two odds for every even:
In[568]:= Table[%, {n, 5}]
Out[568]= {{1, 1/3, -1/2}, {1/5, 1/7, -1/4}, {1/9, 1/11, -1/6}, {1/13, 1/15, -1/8}, {1/17, 1/19, -1/10}}
Adding them all up, In[569]:= Inactive[Sum][Total[%%], {n, \[Infinity]}]
Out[569]= Inactive[Sum][-1/(2*n) + 1/(-3 + 4*n) + 1/(-1 + 4*n), {n, Infinity}]
In[570]:= Activate[%]
Out[570]= Log[8]/2
I.e., 3/2 of the unpermuted series. --rwg
On 2015-09-09 10:03, James Propp wrote:
Can anyone provide a favorite web-reference?
For my upcoming Mathematical Enchantments column, I composed a proof that 1 - 1/2 + 1/3 - 1/4 + ... is both positive and zero, but it really doesn't fit into the (already over-long) article, so I'd like to outsource the job of showing "You can get nonsensical results by rearranging conditionally convergent series" to some website that already exists.
Jim Propp
participants (7)
-
Allan Wechsler -
Dan Asimov -
Dan Asimov -
Eugene Salamin -
James Propp -
Marc LeBrun -
rwg