nLab Mercator series

Idea

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

Statement

Proposition

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

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

Proof

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.

Related concepts

• Chern form

• geometric series

References

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