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In mathematics, the Mercator series or Newton–Mercator series is the
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor se ...
for the
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
: :\ln(1+x)=x-\frac+\frac-\frac+\cdots In summation notation, :\ln(1+x)=\sum_^\infty \frac x^n. The series converges to the natural logarithm (shifted by 1) whenever -1 .


History

The series was discovered independently by
Johannes Hudde Johannes (van Waveren) Hudde (23 April 1628 – 15 April 1704) was a burgomaster (mayor) of Amsterdam between 1672 – 1703, a mathematician and governor of the Dutch East India Company. As a "burgemeester" of Amsterdam he ordered that t ...
and
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the g ...
. It was first published by
Nicholas Mercator Nicholas (Nikolaus) Mercator (c. 1620, Holstein – 1687, Versailles), also known by his German name Kauffmann, was a 17th-century mathematician. He was born in Eutin, Schleswig-Holstein, Germany and educated at Rostock and Leyden after which he ...
, in his 1668 treatise ''Logarithmotechnia''.


Derivation

The series can be obtained from
Taylor's theorem In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the ...
, by inductively computing the ''n''th derivative of \ln(x) at x=1 , starting with :\frac\ln(x)=\frac1. Alternatively, one can start with the finite
geometric series In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each su ...
(t\ne -1) :1-t+t^2-\cdots+(-t)^=\frac which gives :\frac1=1-t+t^2-\cdots+(-t)^+\frac. It follows that :\int_0^x \frac=\int_0^x \left(1-t+t^2-\cdots+(-t)^+\frac\right)\ dt and by termwise integration, :\ln(1+x)=x-\frac+\frac-\cdots+(-1)^\frac+(-1)^n \int_0^x \frac\ dt. If -1 , the remainder term tends to 0 as n\to\infty. This expression may be integrated iteratively ''k'' more times to yield :-xA_k(x)+B_k(x)\ln(1+x)=\sum_^\infty (-1)^\frac, where :A_k(x)=\frac1\sum_^kx^m\sum_^\frac and :B_k(x)=\frac1(1+x)^k are polynomials in ''x''.


Special cases

Setting x=1 in the Mercator series yields the alternating harmonic series :\sum_^\infty \frac=\ln(2).


Complex series

The complex power series :\sum_^\infty \frac=z+\frac+\frac+\frac+\cdots is the
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor se ...
for -\log(1-z) , where log denotes the
principal branch In mathematics, a principal branch is a function which selects one branch ("slice") of a multi-valued function. Most often, this applies to functions defined on the complex plane. Examples Trigonometric inverses Principal branches are use ...
of the
complex logarithm In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related: * A complex logarithm of a nonzero complex number z, defined to b ...
. This series converges precisely for all complex number , z, \le 1,z\ne 1. In fact, as seen by the
ratio test In mathematics, the ratio test is a test (or "criterion") for the convergence of a series :\sum_^\infty a_n, where each term is a real or complex number and is nonzero when is large. The test was first published by Jean le Rond d'Alembert ...
, it has
radius of convergence In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series ...
equal to 1, therefore converges absolutely on every disk ''B''(0, ''r'') with radius ''r'' < 1. Moreover, it converges uniformly on every nibbled disk \overline\setminus B(1,\delta), with ''δ'' > 0. This follows at once from the algebraic identity: :(1-z)\sum_^m \frac=z-\sum_^m \frac-\frac, observing that the right-hand side is uniformly convergent on the whole closed unit disk.


See also

* John Craig


References

* * Anton von Braunmühl (1903
Vorlesungen über Geschichte der Trigonometrie
Seite 134, via
Internet Archive The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music ...
* 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 Walter William Rouse Ball (14 August 1850 – 4 April 1925), known as W. W. Rouse Ball, was a British mathematician, lawyer, and fellow at Trinity College, Cambridge, from 1878 to 1905. He was also a keen amateur magician, and the founding ...
{{DEFAULTSORT:Mercator Series Mathematical series Logarithms