Seeking help in conducting calculations

[Alexander Povolotsky]

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