Mercator series

**analysis** (differential/integral calculus, functional analysis, topology)

metric space, normed vector space

open ball, open subset, neighbourhood

convergence, limit of a sequence

compactness, sequential compactness

continuous metric space valued function on compact metric space is uniformly continuous

…

…

The “Mercator series” (so named after its appearance in Mercator 1667) is the Taylor series of the natural logarithm around 1.

The Taylor series of the natural logarithm around $1 \in \mathbb{R}$ is the following series:

(1)$\begin{aligned}
\underoverset{n = 0}{\infty}{\sum}
\tfrac{1}{n!}
\left(
\frac{d^n}{ d x^n} ln(1 + x)
\right)_{\vert x = 0}
x^n
&
\;\;
=
\;\;
\underoverset{n = 1}{\infty}{\sum}
\frac
{(-1)^{n+1}}
{n}
x^n
\\
&
\;\;
=
\;\;
x
- \tfrac{1}{2} x^2
+ \tfrac{1}{3} x^3
- \tfrac{1}{4} x^4
+ \cdots
\,.
\end{aligned}$

For the first two terms notice that

$ln(1 + x)
\;\xrightarrow{x \to 0}\;
ln(1)
\,=\,
0$

and that the derivative of the natural logarithm is:

$\frac{d}{d x} \ln(1 + x)
\;=\;
\tfrac{1}{1+x}
\;\xrightarrow{ x \to 0 }\;
1
\,.$

From here on, noticing for $k \in \mathbb{N}_+$ that:

$\frac{d}{d x}
\left(
\frac{1}{(1 + x)^k}
\right)
\;=\;
- k
\frac{1}{(1 + x)^{k+1}}
\;\xrightarrow{x \to 0}\;
- k$

we obtain for $n \in \mathbb{N}_+$, by induction:

$\begin{aligned}
\frac{d^n}{d x^n}
ln(1 + x)
&
\;=\;
\frac{d^{n-1}}{d x^{n-1}}
\left(
\frac{1}{1 + x}
\right)
\\
&
\;=\;
(n-1)! \cdot (-1)^{n-1}
\frac{1}{(1 + x)^{n-1}}
\\
&
\;\xrightarrow{ x \to 0 }\;
(n-1)! \cdot (-1)^{n+1}
\end{aligned}
\,.$

Plugging this into the defining equation on the left of (1) and using

$\frac{(n-1)!}{n!} = \frac{1}{n}$

yields the claim.

Apparently first published in:

- Nicholas Mercator,
*Logarithmotechnia: Sive Methodus constuendi Logrithmos*, London 1667 (GoogleBooks)

but will have been known before that.

See also:

- Wikipedia,
*Mercator series*

Last revised on July 25, 2021 at 03:40:01. See the history of this page for a list of all contributions to it.