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User:Jaume Oliver Lafont

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

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))
${\displaystyle {\frac {\sqrt {b}}{2}}\log {\frac {{\sqrt {b}}+1}{{\sqrt {b}}-1}}=\sum _{k=0}^{\infty }{\frac {1}{(2k+1)b^{k}}}}$

Permutations of integers

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

Table of logarithms

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

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

Equivalently, as permutations of the harmonic series minus itself:

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

Riemann series theorem makes this possible.

Series for ${\displaystyle e^{\pi }}$

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

Approximations to ${\displaystyle e}$ and ${\displaystyle \pi }$

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

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

${\displaystyle \pi \approx 4-\left({\dfrac {2}{5}}\right)^{\frac {1}{6}}=3.141(6)}$

References

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