The natural logarithm of 2 is a transcendental quantity that arises often in decay problems, especially when half-lives are being
converted to decay constants. has numerical value
converges to the natural logarithm of 2, where is the Dirichlet
eta function. This identity follows immediately from setting in the Mercator series,
yielding
(Borwein and Bailey 2002, p. 50), where is a tangent number. This
means that truncating the series for at half a large power of 10 can give a decimal expansion
for
whose decimal digits are largely correct, but where wrong digits occur with precise
regularity.
For example, taking gives a decimal value equal to the second row
of digits above, where the sequence of differences from the decimal digits of in the top row is precisely the tangent numbers with alternating signs (Borwein and
Bailey 2002, p. 49).
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