Mercator series explained

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) + …

In summation notation,

ln ( 1+ x) = \sum ^{inf ty} _{n =1} (( -1 ) ^{n+1})/(n) x ^{n.}

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

-1 <x\le 1 .

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

The series can be obtained from Taylor's theorem, by inductively computing the n^th derivative of

ln (x) at x =1 , starting with(d)/(ln (x) = dx) (1 {x}.)/()

Alternatively, one can start with the finite geometric series (

t\ne -1 )1- t + t ^{2- … +(-t)} ^{n-1} = (1- ( - t) ^{n})/(1+ t)

which gives

(1 { 1+ t} =1- t + t)/(^{2- … +(-t)}) ^{n-1} + (( - t) ^{n})/(1+ t) .

It follows that

\int ^{x} _{0} (dt)/(1+ t) = \int ^{x} _{0} \left( 1- t + t ^{2- … +(-t)} ^{n-1} + (( - t) ^{n})/(1+ t) \right) dt

and by termwise integration,

ln ( 1+ x) = x - (x ^{2})/(+ 2) (x ^{3})/(3) - … + ( -1 ) ^{n-1} (x ^{n})/(n) + ( -1 ) ^{n} \int ^{x} _{0} (t ^{n})/(1+ t) dt.

If

-1 <x\le 1 , the remainder term tends to 0 as n\to inf ty .

This expression may be integrated iteratively k more times to yield

- xA _{k(x)+B} _{k(x)ln(1+x)=\sum} ^{inf ty} ^{n =1} ( -1 ) ^{n-1} (x ^{n+k})/(n(n +1 ) … (n + k)) , whereA ^{k{k\choose} _{k(x) = (1 {k!}\sum)/(_{m=0})} m}x ^{k - m} _{l =1} (( - x) ^{l-1})/(l) andB _{k(x) = (1 {k!}( 1+ x))/(^{k})} are polynomials in x.^[3]

Special cases

Setting

x =1 in the Mercator series yields the alternating harmonic series\sum ^{inf ty} _{k =1} (( -1 ) ^{k+1})/(k) =ln ( 2 ).

Complex series

The complex power series

\sum ^{inf ty} _{n =1} (z ^{n})/(= z + n) (z ^{2})/(+ 2) (z ^{3})/(+ 3) (z ^{4})/(4) + …

is the Taylor series for

-log ( 1- z) , where log denotes the principal branch of the complex logarithm. This series converges precisely for all complex number |z|\le 1 ,z\ne 1 . 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 $\backslash overline\backslash setminus\; B(1,\backslash delta)$, with δ > 0. This follows at once from the algebraic identity: ( 1- z)\sum ^{m} _{n =1} (z ^{n})/(n) = z - \sum ^{m} _{n =2} (z ^{n})/(- n(n -1 )) (z ^{m+1})/(m) , observing that the right-hand side is uniformly convergent on the whole closed unit disk.

See also

• John Craig
• Natural logarithm plus 1

References

• • 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

Notes and References

1. Vermij . Rienk . Bijdrage tot de bio-bibliografie van Johannes Hudde . Gewina / TGGNWT . 3 February 2012 . 18 . 1 . 25–35 . 1874/251283 . nl . 0928-303X.
2. Book: Roy, Ranjan . 2021 . 1st ed. 2011 . Series and Products in the Development of Mathematics . 2nd . 1 . Cambridge University Press . 107, 167.
3. Luis A.. Medina. Victor H.. Moll . Victor Hugo Moll . Eric S.. Rowland. Iterated primitives of logarithmic powers. 0911.1325. 2011. 10.1142/S179304211100423X. 7. International Journal of Number Theory. 3 . 623–634. 115164019 .