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Mercator series
Taylor series for the natural logarithm From Wikipedia, the free encyclopedia
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In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm:

The series converges to the natural logarithm (shifted by 1) whenever .
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History
The series was discovered independently by Johannes Hudde (1656)[1] and Isaac Newton (1665) but neither published the result. Nicholas Mercator also independently discovered it, and included values of the series for small values in his 1668 treatise Logarithmotechnia; the general series was included in John Wallis's 1668 review of the book in the Philosophical Transactions.[2]
Derivation
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The series can be obtained by computing the Taylor series of at :
and substituting all with . Alternatively, one can start with the finite geometric series ()
which gives
It follows that
and by termwise integration,
If , the remainder term tends to 0 as .
This expression may be integrated iteratively k more times to yield
where
and
are polynomials in x.[3]
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Special cases
Setting in the Mercator series yields the alternating harmonic series
Complex series
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The complex power series
is the Taylor series for , where log denotes the principal branch of the complex logarithm. This series converges precisely for all complex number . In fact, as seen by the ratio test, it has radius of convergence equal to 1, therefore converges absolutely on every disk B(0, r) with radius r < 1. Moreover, it converges uniformly on every nibbled disk , with δ > 0. This follows at once from the algebraic identity:
observing that the right-hand side is uniformly convergent on the whole closed unit disk.
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See also
References
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