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User:Jaume Oliver Lafont

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Engineer. Started contributing to the OEIS in 2007, in sequence A058962.

Contents

BBP formulas

  • A154920 Denominators of a ternary BBP-type formula for log(3)
  • A165998 Denominators of Taylor series expansion of 1/(3*x)*log((1+x)/(1-x)^2)
  • A164985 Denominators of ternary BBP-type series for log(5)
  • A166486 Periodic sequence [0,1,1,1] of length 4
  • A165132 Primes whose logarithms are known to possess ternary BBP formulas
  • P(1,b,2,(1,0))
\frac{\sqrt{b}}{2}\log{\frac{\sqrt{b}+1}{\sqrt{b}-1}} = \sum_{k=0}^\infty \frac{1}{(2k+1)b^k}

Other binary BBP formulas

Permutations of integers

  • A166711 Sum_{k>0} 1/a(k) = log(2)
  • A166871 Sum_{k>0} 1/a(k) = log(3/2)

General expressions for log(p/q) appear in the sequences.

Table of logarithms

As generalized Mercator series (or BBP-type formulas in base 1):

log(1) = \left(\frac{0}{1}\right) + \left(\frac{0}{2}\right) +...
log(2) = \left(\frac{1}{1} -\frac{1}{2}\right) + \left(\frac{1}{3} -\frac{1}{4}\right) +...
log(3) = \left(\frac{1}{1}+\frac{1}{2} -\frac{2}{3}\right) + \left(\frac{1}{4}+\frac{1}{5} -\frac{2}{6}\right) +...
log(4) = \left(\frac{1}{1}+\frac{1}{2}+\frac{1}{3} -\frac{3}{4}\right) + \left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7} -\frac{3}{8}\right) +...
log(5) = \left(\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4} -\frac{4}{5}\right) + \left(\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9} -\frac{4}{10}\right) +...
\dots

Equivalently, as permutations of the harmonic series minus itself:

log(1) = \left(\frac{1}{1}-\frac{1}{1}\right) + \left(\frac{1}{2}-\frac{1}{2}\right) +...
log(2) = \left(\frac{1}{1}+\frac{1}{2} -\frac{1}{1}\right) + \left(\frac{1}{3}+\frac{1}{4} -\frac{1}{2}\right) +...
log(3) = \left(\frac{1}{1}+\frac{1}{2}+\frac{1}{3} -\frac{1}{1}\right) + \left(\frac{1}{4}+\frac{1}{5}+\frac{1}{6} -\frac{1}{2}\right) +...
log(4) = \left(\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4} -\frac{1}{1}\right) + \left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8} -\frac{1}{2}\right) +...
log(5) = \left(\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5} -\frac{1}{1}\right) + \left(\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10} -\frac{1}{2}\right) +...
\dots

Riemann series theorem makes this possible.

Series for eπ


e^{\pi}-\pi=1+\sum_{k=0}^\infty\frac{\Gamma(\frac{1}{2})^{2k+4}}{\Gamma(k+3)}

Approximations to e and π

e \approx \left(\sum_{k=1}^8 \frac{1}{k}\right) \left(1+\frac{1}{80^2}\right) =2.7182818(080)

\pi \approx \left(\sum_{k=1}^{12} \frac{1}{k}\right) \left(1+\frac{1}{9^2}\right)=3.1415(219)

Rational recurrences

Lucas numbers and log(2)

References

Some of these results have been included in the following works.

Notebook

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