In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm:
ln ( 1 + x ) = x − x 2 2 + x 3 3 − x 4 4 + ⋯ {\displaystyle \ln(1+x)=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}+\cdots }
In summation notation,
ln ( 1 + x ) = ∑ n = 1 ∞ ( − 1 ) n + 1 n x n . {\displaystyle \ln(1+x)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}x^{n}.}
The series converges to the natural logarithm (shifted by 1) whenever − 1 < x ≤ 1 {\displaystyle -1<x\leq 1} .
History The series was discovered independently by Johannes Hudde (1656) 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.
Derivation The series can be obtained by computing the Taylor series of ln ( x ) {\displaystyle \ln(x)} at x = 1 {\displaystyle x=1} :
ln ( x ) = ( x − 1 ) − ( x − 1 ) 2 2 + ( x − 1 ) 3 3 − ⋯ , {\displaystyle \ln(x)=(x-1)-{\frac {(x-1)^{2}}{2}}+{\frac {(x-1)^{3}}{3}}-\cdots ,}
and substituting all x {\displaystyle x} with x + 1 {\displaystyle x+1} . Alternatively, one can start with the finite geometric series ( t ≠ − 1 {\displaystyle t\neq -1} )
1 − t + t 2 − ⋯ + ( − t ) n − 1 = 1 − ( − t ) n 1 + t {\displaystyle 1-t+t^{2}-\cdots +(-t)^{n-1}={\frac {1-(-t)^{n}}{1+t}}}
which gives
1 1 + t = 1 − t + t 2 − ⋯ + ( − t ) n − 1 + ( − t ) n 1 + t . {\displaystyle {\frac {1}{1+t}}=1-t+t^{2}-\cdots +(-t)^{n-1}+{\frac {(-t)^{n}}{1+t}}.}
It follows that
∫ 0 x d t 1 + t = ∫ 0 x ( 1 − t + t 2 − ⋯ + ( − t ) n − 1 + ( − t ) n 1 + t ) d t {\displaystyle \int _{0}^{x}{\frac {dt}{1+t}}=\int _{0}^{x}\left(1-t+t^{2}-\cdots +(-t)^{n-1}+{\frac {(-t)^{n}}{1+t}}\right)\ dt}
and by termwise integration,
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