[seqfan] Re: A series for exp(Pi)

[Alexander Povolotsky]

Also paragraph 4.3 in
http://www.physik.uni-augsburg.de/theo1/hanggi/Papers/24.pdf

could be of interest for the subject matter.

On 10/26/09, Alexander Povolotsky <apovolot at gmail.com> wrote:
> Also another another observation is (as I already posted in
> http://groups.google.com/group/sci.math.symbolic/browse_thread/thread/1b5e3b73f1d93b8d/88d5669fc28f3bf8?hl=en#88d5669fc28f3bf8
> discussion)
>
> sum(1/((3*2^(-3 + n)*gamma(((-1 - 6*i) + n)/2)*gamma(((-1 + 6*i) +
> n)/2)*((1 + (-1)^n)*csch(3Pi) >- (-1 +
> (-1)^n)*sech(3*Pi))*sinh(6*Pi))/Pi*2),n=0...infinity)
>
> gives the real value 23.14... - which is very close to exp(Pi) but NOT
> exactly ....
>
>> Maybe some of these approximations helps give some clue why exp(Pi)-Pi
>> is almost integer. Are there two rational approximations for exp(Pi)
>> and Pi whose difference is 20?
>
> Perhaps also exp(Pi)-Pi almost integer phenomena could be related to
> Ramanujan's almost integers - see discussion at
> http://groups.google.com/group/sci.math.research/browse_thread/thread/3d24137c9a860893/d9a76edce395b414?hl=en#d9a76edce395b414
>
>
>
>
>
> On 10/26/09, Jaume Oliver i Lafont <joliverlafont at gmail.com> wrote:
>> Thanks again, -Alexander, Richard, Maximilian-, for your work.
>>
>> with x=1/2, exp(6asin(x)) computes exp(Pi),
>> and with x=-1/2, it computes its reciprocal 1/exp(Pi)
>>
>> exp(Pi) as computed by plugging 1/2, one more term each line.
>> a(n)=if(n<2,[1,6][n+1],((n-2)^2+36)*a(n-2))
>> b(n)=if(!n,1,-2*n*b(n-1))
>> s=0; for (n=0,25,s+=a(n)/b(n);print(s*1.))
>>
>> 1.000000000000000000000000000
>> 4.000000000000000000000000000
>> 8.500000000000000000000000000
>> 13.12500000000000000000000000
>> 16.87500000000000000000000000
>> 19.47656250000000000000000000
>> 21.10156250000000000000000000
>> 22.04617745535714285714285714
>> 22.56849888392857142857142857
>> 22.84729149228050595238095238
>> 22.99238077799479166666666667
>> 23.06651426703382880140692641
>> 23.10388574971781148538961039
>> 23.12253792564590255935470779
>> 23.13177812740842575044932746
>> 23.13633014653373369111938099
>> 23.13856319529301012896724741
>> 23.13965517782215109635225106
>> 23.14018789860459286093687279
>> 23.14044732427854813462117556
>> 23.14057349499017907886490176
>> 23.14063479974765541437244235
>> 23.14066456729650340771133013
>> 23.14067901510743038994309736
>> 23.14068602558089096808450209
>> 23.14068942683638002848489729
>>
>> exp(Pi) as computed by plugging -1/2 and then inverting the result
>> // a(n) the same as before
>> b(n)=if(!n,1,-2*n*b(n-1))
>> s=0; for (n=0,25,s+=a(n)/b(n);print(1./s))
>> 1.000000000000000000000000000
>> -0.5000000000000000000000000000
>> 0.4000000000000000000000000000
>> -0.4705882352941176470588235294
>> 0.6153846153846153846153846154
>> -1.024000000000000000000000000
>> 1.542168674698795180722891566
>> -3.376354215732454074422986340
>> 4.421961752004935225169648365
>> -18.99384471003891909796019763
>> 10.81775164672776721007372939
>> 54.62343236258507389984620172
>> 17.96020743645145456136266233
>> 27.00770681323076388733446709
>> 21.61383126966584175758151264
>> 23.97239285691561645574523350
>> 22.75431949918253546357694280
>> 23.33410957158627851806014370
>> 23.04761469650443776683304108
>> 23.18624859190561371987319442
>> 23.11861680114008971589106302
>> 23.15142888596491787456660092
>> 23.13548480540554593992088942
>> 23.14322059144052715847967643
>> 23.13946633025879637420577321
>> 23.14128762450017775606900208
>>
>> Maybe some of these approximations helps give some clue why exp(Pi)-Pi
>> is almost integer. Are there two rational approximations for exp(Pi)
>> and Pi whose difference is 20?
>>
>>
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