Mercator series

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

In summation notation,

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

History[edit]

The series was discovered independently by Nicholas Mercator, Isaac Newton and Gregory Saint-Vincent. It was first published by Mercator, in his 1668 treatise Logarithmotechnia.

Derivation[edit]

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.[1]

Special cases[edit]

Setting in the Mercator series yields the alternating harmonic series

Complex series[edit]

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.

References[edit]

  1. ^ Medina, Luis A.; Moll, Victor H.; Rowland, Eric S. (2009). "Iterated primitives of logarithmic powers". arXiv:0911.1325.