# Newton-Mercator Series/Example/2

 It has been suggested that this page or section be merged into Alternating Harmonic Series sums to ln 2. (Discuss)

## Example of use of the Newton-Mercator Series

$\ln 2 = 1 - \dfrac 1 2 + \dfrac 1 3 - \dfrac 1 4 + \dfrac 1 5 - \cdots$

## Proof

From the Newton-Mercator Series:

 $\displaystyle \map \ln {1 + x}$ $=$ $\displaystyle x - \dfrac {x^2} 2 + \dfrac {x^3} 3 - \dfrac {x^4} 4 + \cdots$ $\displaystyle$ $=$ $\displaystyle \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n x^n$

This is valid for $-1 < x \le 1$.

Setting $x = 1$:

 $\displaystyle \map \ln {1 + 1}$ $=$ $\displaystyle 1 - \dfrac {1^2} 2 + \dfrac {1^3} 3 - \dfrac {1^4} 4 + \cdots$ $\displaystyle \leadsto \ \$ $\displaystyle \ln 2$ $=$ $\displaystyle 1 - \dfrac 1 2 + \dfrac 1 3 - \dfrac 1 4 + \dfrac 1 5 - \cdots$

$\blacksquare$