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Encyclopedia > Mercator series

In mathematics, the Mercator series or Newton-Mercator series is the Taylor series for the natural logarithm. It is given by Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... As the degree of the Taylor series rises, it approaches the correct function. ... The natural logarithm, invented by John Napier, is the logarithm to the base e, where e is equal to 2. ...

ln(1+x)=sum_{n=1}^infty frac{(-1)^{n+1}}{n} x^n = x - frac{x^2}{2} + frac{x^3}{3} - frac{x^4}{4} + ldots,

valid for -1 < x le 1.

Contents


History

The series was discovered independently by Isaac Newton, Nicholas Mercator and Gregory Saint-Vincent. It was first published by Mercator, in his 1668 treatise Logarithmo-technica. Sir Isaac Newton, PRS, (4 January [O.S. 25 December 1642] 1643 – 31 March [O.S. 20 March] 1727) was an English physicist, mathematician, astronomer, alchemist, inventor and natural philosopher who is generally regarded as one of the most influential scientists in history. ... Nicholas (Nikolaus) Mercator (c. ...


Derivation

The series can be derived by repeatedly differentiating the natural logarithm, starting with

frac{d}{dx} ln x = frac{1}{x}.

Alternatively, one can start with the geometric series (t neq -1) In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ...

1 - t + t^2 - ldots + (-t)^{n-1} = frac{1 - (-t)^n}{1+t}

which gives

frac{1}{1+t} = 1 - t + t^2 - ldots + (-t)^{n-1} + frac{(-t)^n}{1+t}.

It follows that

int_0^x frac{dt}{1+t} = int_0^x left( 1 - t + t^2 - ldots + (-t)^{n-1} + frac{(-t)^n}{1+t} right) dt

and by termwise integration,

ln(1+x) = x - frac{x^2}{2} + frac{x^3}{3} - ldots + (-1)^{n-1}frac{x^n}{n} + (-1)^n int_0^x frac{t^n}{1+t} dt.

If -1 < x le 1, the remainder term vanishes when n to infty.


Special cases

Setting x = 1, the Mercator series reduces to the alternating harmonic series

sum_{k = 1}^infty frac{(-1)^{k + 1}}{k} = ln 2.

References

  • Eric W. Weisstein, Mercator Series at MathWorld.
  • 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

 

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