Definition:Newton-Mercator Series

Theorem
Let $\ln x$ denote the natural logarithm function.

Then:

The series converges to the natural logarithm (shifted by $1$) for $-1 < x \le 1$.

This is known as the Newton-Mercator series.

Proof
From Sum of Infinite Geometric Progression, we know that:


 * $\displaystyle \sum_{n \mathop = 0}^\infty x^n$ converges to $\dfrac 1 {1 - x}$

for $\size x < 1$

which implies that:


 * $\displaystyle \sum_{n \mathop = 0}^\infty (-1)^n x^n$ converges to $\dfrac 1 {1 + x}$

We also know from Definition:Natural Logarithm that:


 * $\map \ln {x + 1} = \displaystyle \int_0^x \frac {\d t} {1 + t}$

Combining these facts, we get:


 * $\map \ln {x + 1} = \displaystyle \int_0^x \sum_{n \mathop = 0}^\infty \paren {-1}^n t^n \rd t$

From Linear Combination of Integrals, we can rearrange this to:


 * $\displaystyle \sum_{n \mathop = 0}^\infty \paren {-1}^n \displaystyle \int_0^x t^n \rd t$

Then, using Integral of Power:


 * $\displaystyle \sum_{n \mathop = 0}^\infty \dfrac {\paren {-1}^n} {n + 1} x^{n + 1}$

We can shift $n + 1$ into $n$:


 * $\displaystyle \sum_{n \mathop = 1}^\infty \dfrac {\paren {-1}^{n - 1} } n x^n$

This is equivalent to:


 * $\displaystyle \sum_{n \mathop = 1}^\infty \dfrac {\paren {-1}^{n + 1} } n x^n$

Finally, we check the bounds $x = 1$ and $x = -1$.

For $x = -1$, we get:


 * $\displaystyle \sum_{n \mathop = 1}^\infty \dfrac {\paren {-1}^{n + 1} } n \paren {-1}^n$

$\paren {-1}^{n + 1}$ and $\paren {-1}^n$ will always have different signs, which implies their product will be $-1$.

This means we get:


 * $-\displaystyle \sum_{n \mathop = 1}^\infty \dfrac 1 n$

This is the harmonic series which we know to be divergent.

We then check $x = 1$.

We get:


 * $\displaystyle \sum_{n \mathop = 1}^\infty \dfrac {\paren {-1}^{n + 1} } n$

This is the alternating harmonic series which we know to be convergent.

Therefore, we can conclude that:
 * $\map \ln {x + 1} = \displaystyle \sum_{n \mathop = 1}^\infty \dfrac {\paren {-1}^{n + 1} } n x^n$ for $-1 < x \le 1$.

Also known as
The Newton-Mercator series is also known as the Mercator series, or Mercator's series.