Mercator series

Summary

In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm:

Polynomial approximation to logarithm with n=1, 2, 3, and 10 in the interval (0,2).

In summation notation,

The series converges to the natural logarithm (shifted by 1) whenever .

History

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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 from Taylor's theorem, by inductively computing the nth derivative of   at   , starting 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]

Special cases

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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.

See also

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References

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  1. ^ Vermij, Rienk (3 February 2012). "Bijdrage tot de bio-bibliografie van Johannes Hudde". GEWINA / TGGNWT (in Dutch). 18 (1): 25–35. ISSN 0928-303X.
  2. ^ Roy, Ranjan (2021) [1st ed. 2011]. Series and Products in the Development of Mathematics. Vol. 1 (2nd ed.). Cambridge University Press. pp. 107, 167.
  3. ^ Medina, Luis A.; Moll, Victor H.; Rowland, Eric S. (2011). "Iterated primitives of logarithmic powers". International Journal of Number Theory. 7 (3): 623–634. arXiv:0911.1325. doi:10.1142/S179304211100423X. S2CID 115164019.
  • Weisstein, Eric W. "Mercator Series". MathWorld.
  • Anton von Braunmühl (1903) Vorlesungen über Geschichte der Trigonometrie, Seite 134, via Internet Archive
  • Eriksson, Larsson & Wahde. Matematisk analys med tillämpningar, part 3. Gothenburg 2002. p. 10.
  • Some Contemporaries of Descartes, Fermat, Pascal and Huygens from A Short Account of the History of Mathematics (4th edition, 1908) by W. W. Rouse Ball