Seeking help in conducting calculations

[Alexander Povolotsky]

 The same seems to be happening

when constructing c.f. from Engel's coefficients for e-1  (1.718...)

e - 1 = (1/1 + 1/1*2 + 1/1*2*3 ... + 1/n! + ...)

1/(1+1)= 1/2=0.5
1/(1+1/(1+2)) = 3/4=0.75
1/(1+1/(1+2/(1+6))) = 9/16= 0.5625
1/(1+1/(1+2/(1+6/(1+24)))) = 81/112=0.723214286
1/(1+1/(1+2/(1+6/(1+24/(1+120))))) = 1161/2032=0.571358268
1/(1+1/(1+2/(1+6/(1+24/(1+120/(1+720)))))) = 59481/82672=0.719481808

Thanks,
Best Regards,
Alexander Povolotsky
-------------------------------------------
On Feb 19, 2008 7:24 PM, Alexander Povolotsky <apovolot at gmail.com> wrote:
> Hi,
>
> Below are my  partial results when I tried to use OEIS's A006784 "Pi -
> 3" Engel's expansion coefficients:
>
> 8, 8, 17, 19, 300, 1991, 2492, 7236, 10586, 34588, ...
>
>  to form a continued fraction expansion and to see to what value it comes to.
>
> I noticed that after few steps the results are showing two
> sequentially interchanging (and possibly slowly converging to each
> other) trends:
>
> 0.529411765->0.517267552->0.508117341->0.5021273->...
>
> and
>
> 0.152941176->0.158526794->0.16041072->0.160983247->...
>
> My old Pentium PC at home is too slow and low on memory to find out
> whether there is convergence and what the value of such possible
> convergent limit will be.
> Could someone please help with conducting calculations ?
>
> 1/(1+8)=1/9=0.111111111
>
> 1/(1+8/(1+8))=9/17=0.529411765
>
> 1/(1+8/(1+8/(1+17)))=13/85=0.152941176
>
> 1/(1+8/(1+8/(1+17/(1+19))))=197/493=0.39959432
>
> 1/(1+8/(1+8/(1+17/(1+300))))=1363/2635=0.517267552
>
> 1/(1+8/(1+8/(1+17/(1+300/(1+1991)))))=4541/28645=0.158526794
>
> 1/(1+8/(1+8/(1+17/(1+300/(1+1991/(1+2492))))))=1711921/3369145=0.508117341
>
> 1/(1+8/(1+8/(1+17/(1+300/(1+1991/(1+2492/(1+7236)))))))=100287949/625194805=
> 0.16041072
>
> 1/(1+8/(1+8/(1+17/(1+300/(1+1991/(1+2492/(1+7236/(1+10586))))))))=110440507/219945235=
> 0.5021273
>
> 1/(1+8/(1+8/(1+17/(1+300/(1+1991/(1+2492/(1+7236/(1+10586/(1+34588)))))))))=498140323381/3
> 094361268445= 0.160983247
>
> Thanks,
> Best Regards,
> Alexander Povolotsky
>