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

[Alexander Povolotsky]

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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