# [seqfan] A sequence of BBP formulas for ln(3) in base 9^k?

Jaume Oliver i Lafont joliverlafont at gmail.com
Tue Dec 23 17:29:58 CET 2008

Hello seqfans,

Alexander Povolotsky sent to this list and to
http://research.att.com/~njas/sequences/A002391 the formula:

ln(3) = 1/4*(1+ Sum((1/(9)^(k+1))*(27/(2*k+1) + 4/(2*k+2) +
1/(2*k+3)), k = 0 .. infinity) )

After some manipulation, it can be arranged to:

ln(3) = Sum((1/9)^(k+1)*(9/(2*k+1) + 1/(2*k+2)), k = 0 .. infinity) ),

exploiting the fact that 3 and 1 are congruent mod 2.

In the notation used in
http://crd.lbl.gov/~dhbailey/dhbpapers/bbp-formulas.pdf this is
expressed as:

ln(3)=(1/9)P(1,9,2,(9,1))

At http://mathworld.wolfram.com/BBP-TypeFormula.html there is
ln(3)=(1/729)P(1,729,6,(729,81,81,9,9,1))

"Between" these two formulas another one can be "interpolated", which
has been experimentally checked:

ln(3)=(1/81)P(1,81,4,(81,9,9,1))

and "beyond" the one at Mathworld we also devise and succesfully check

ln(3)=(1/6561)P(1,6561,8,(6561,729,729,81,81,9,9,1))

This suggests an infinite sequence of BBP formulas for ln(3) in base
9^k, which would be easy to continue.

Regards,
Jaume