Editing List of logarithmic identities (section)

=== Series representation ===

The natural logarithm <math>\ln(1 + x)</math> has a well-known [[Taylor series]]<ref>{{cite web
 | last = Weisstein
 | first = Eric W.
 | title = Mercator Series
 | website = MathWorld--A Wolfram Web Resource
 | url = https://mathworld.wolfram.com/MercatorSeries.html
 | access-date = 2024-04-24
}}</ref> expansion that converges for <math>x</math> in the [[Interval (mathematics)|open-closed interval]] {{open-closed|−1, 1}}:

<math display="block">\ln(1 + x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \frac{x^6}{6} + \cdots.</math>

Within this interval, for <math>x = 1</math>, the series is [[Conditional convergence|conditionally convergent]], and for all other values, it is [[Absolute convergence|absolutely convergent]]. For <math>x > 1</math> or <math>x \leq -1</math>, the series does not converge to <math>\ln(1 + x)</math>. In these cases, different representations<ref>To extend the utility of the Mercator series beyond its conventional bounds one can calculate <math display="inline"> \ln(1 + x)</math> for <math display="inline"> x = -\frac{n}{n+1}</math> and <math display="inline"> n \geq 0</math> and then [[#Using_simpler_operations|negate the result]], <math display="inline"> \ln\left(\frac{1}{n+1}\right)</math>, to derive <math display="inline"> \ln(n + 1)</math>. [[Natural_logarithm_of_2#Binary_rising_constant_factorial|For example]], setting <math display="inline"> x = -\frac{1}{2}</math> yields <math display="inline"> \ln 2 = \sum_{n=1}^{\infty} \frac{1}{n 2^n}</math>.</ref> or methods must be used to evaluate the logarithm.