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 ) = ∑ n = 1 ∞ ( − 1 ) n + 1 n x n . The series converges to the natural logarithm (shifted by 1) whenever − 1 < x ≤ 1 .
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.
The series can be obtained from Taylor's theorem, by inductively computing the nth derivative of ln ( x ) at x = 1 , starting with
d d x ln ( x ) = 1 x . Alternatively, one can start with the finite geometric series ( t ≠ − 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
∫ 0 x d t 1 + t = ∫ 0 x ( 1 − t + t 2 − ⋯ + ( − t ) n − 1 + ( − t ) n 1 + t ) d t and by termwise integration,
ln ( 1 + x ) = x − x 2 2 + x 3 3 − ⋯ + ( − 1 ) n − 1 x n n + ( − 1 ) n ∫ 0 x t n 1 + t d t . If − 1 < x ≤ 1 , the remainder term tends to 0 as n → ∞ .
This expression may be integrated iteratively k more times to yield
− x A k ( x ) + B k ( x ) ln ( 1 + x ) = ∑ n = 1 ∞ ( − 1 ) n − 1 x n + k n ( n + 1 ) ⋯ ( n + k ) , where
A k ( x ) = 1 k ! ∑ m = 0 k ( k m ) x m ∑ l = 1 k − m ( − x ) l − 1 l and
B k ( x ) = 1 k ! ( 1 + x ) k are polynomials in x.
Setting x = 1 in the Mercator series yields the alternating harmonic series
∑ k = 1 ∞ ( − 1 ) k + 1 k = ln ( 2 ) . The complex power series
∑ n = 1 ∞ z n n = z + 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 | ≤ 1 , z ≠ 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 B ( 0 , 1 ) ¯ ∖ B ( 1 , δ ) , with δ > 0. This follows at once from the algebraic identity:
( 1 − z ) ∑ n = 1 m z n n = z − ∑ n = 2 m z n n ( n − 1 ) − z m + 1 m , observing that the right-hand side is uniformly convergent on the whole closed unit disk.
