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# Natural logarithm of 2

(Redirected from Alternating harmonic sequence)

The natural logarithm of 2 has decimal expansion

0.693147180559945309417232121458176568075500134360255254120680009493393...

A002162 Decimal expansion of natural logarithm of 2.

{6, 9, 3, 1, 4, 7, 1, 8, 0, 5, 5, 9, 9, 4, 5, 3, 0, 9, 4, 1, 7, 2, 3, 2, 1, 2, 1, 4, 5, 8, 1, 7, 6, 5, 6, 8, 0, 7, 5, 5, 0, 0, 1, 3, 4, 3, 6, 0, 2, 5, 5, 2, 5, 4, 1, 2, 0, ...}

## Series

From the Maclaurin series expansions of ${\displaystyle \scriptstyle \log(1+x)}$, we have the alternating harmonic series

${\displaystyle \log(2)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}.}$

## Continued fraction

The continued fraction for ${\displaystyle \log(2)}$ is

${\displaystyle \log(2)=0+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{3+{\cfrac {1}{1+{\cfrac {1}{6+{\cfrac {1}{3+{\cfrac {1}{1+{\cfrac {1}{\ddots }}}}}}}}}}}}}}}}}$

A016730 Continued fraction for ln(2).

{0, 1, 2, 3, 1, 6, 3, 1, 1, 2, 1, 1, 1, 1, 3, 10, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 3, 1, 13, 7, 4, 1, 1, 1, 7, 2, 4, 1, 1, 2, 5, 14, 1, 10, 1, 4, 2, 18, 3, 1, 4, 1, 6, 2, 7, 3, 3, ...}