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