nLab
Mercator series

Contents

Contents

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 11 \in \mathbb{R} is the following series:

(1)n=01n!(d ndx nln(1+x)) |x=0x n =n=1(1) n+1nx n =x12x 2+13x 314x 4+. \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)x0ln(1)=0 ln(1 + x) \;\xrightarrow{x \to 0}\; ln(1) \,=\, 0

and that the derivative of the natural logarithm is:

ddxln(1+x)=11+xx01. \frac{d}{d x} \ln(1 + x) \;=\; \tfrac{1}{1+x} \;\xrightarrow{ x \to 0 }\; 1 \,.

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

ddx(1(1+x) k)=k1(1+x) k+1x0k \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 +n \in \mathbb{N}_+, by induction:

d ndx nln(1+x) =d n1dx n1(11+x) =(n1)!(1) n11(1+x) n1 x0(n1)!(1) n+1. \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

(n1)!n!=1n \frac{(n-1)!}{n!} = \frac{1}{n}

yields the claim.

References

Apparently first published in:

but will have been known before that.

See also:

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