logarithm series

The derivative of  ln(1+x)  is 11+x, which can be represented as the sum of geometric seriesMathworldPlanetmath:

11+x= 1-x+x2-x3+-  for-1<x<1.

Integrating both from 0 to x gives

ln(1+x)=x-x22+x33-x44+-  for-1<x<1. (1)

which is valid on the whole open intervalDlmfPlanetmath of convergence-1<x<1  of this power seriesMathworldPlanetmath and in for  x=1, as one may prove.

Replacing x with -x in (1) yields the series

ln(1-x)=-x-x22-x33-x44-  for-1<x<1. (2)

Subtracting (2) from (1) gives

ln1+x1-x= 2(x+x33+x55+x77+) (3)

which also is true for  -1<x<1.  Here the inner function of the logarithmMathworldPlanetmath attains all positive real values when  0<x<1 (its graph (http://planetmath.org/Graph2) is a hyperbola (http://planetmath.org/Hyperbola2) with asymptotes (http://planetmath.org/AsymptotesOfGraphOfRationalFunction)  x=1  and  y=-1).  Thus, in principle, the series (3) can be used for calculating any values of natural logarithmMathworldPlanetmathPlanetmath (http://planetmath.org/NaturalLogarithm2).  For this purpose, one could denote


which implies


and accordingly

lnt= 2[t-1t+1+13(t-1t+1)3+15(t-1t+1)5+]. (4)

For example,

ln3= 2(12+1323+1525+).

The convergence of (4) is the slower the greater is t.

Title logarithm series
Canonical name LogarithmSeries
Date of creation 2013-03-22 18:56:13
Last modified on 2013-03-22 18:56:13
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Topic
Classification msc 33B10
Related topic TaylorSeriesOfArcusSine
Related topic TaylorSeriesOfArcusTangent