Natural Logarithm - Derivative, Taylor Series

Derivative, Taylor Series

The derivative of the natural logarithm is given by

This leads to the Taylor series for ln(1 + x) around 0; also known as the Mercator series

(Leonhard Euler nevertheless boldly applied this series to x= -1, in order to show that the harmonic series equals the (natural) logarithm of 1/(1-1), that is the logarithm of infinity. Nowadays, more formally but perhaps less vividly, we prove that the harmonic series truncated at N is close to the logarithm of N, when N is large).

At right is a picture of ln(1 + x) and some of its Taylor polynomials around 0. These approximations converge to the function only in the region −1 < x ≤ 1; outside of this region the higher-degree Taylor polynomials are worse approximations for the function.

Substituting x − 1 for x, we obtain an alternative form for ln(x) itself, namely

By using the Euler transform on the Mercator series, one obtains the following, which is valid for any x with absolute value greater than 1:

This series is similar to a BBP-type formula.

Also note that is its own inverse function, so to yield the natural logarithm of a certain number y, simply put in for x.