Definition:Newton-Mercator Series

Theorem
Let $\ln x$ be 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
We define $f(x)=\ln(x+1)$

From Nth Derivative of Natural Logarithm, we know that

$f^{(n)}(x)=\dfrac{(n-1)!(-1)^{n-1}}{(x+1)^n}$

and for $x=0$

$f^{(n)}(0)=(n-1)!(-1)^{n-1}$

By Definition:Taylor Series, we get that

$f(x)=\displaystyle \sum_{n \mathop = 0}^\infty \frac {\left({x - \xi}\right)^n} {n!} f^{\left({n}\right)} \left({\xi}\right)$

Plugging in $\xi=0$ and our initial statement for $f^{(n)}(0)$, we get

$\displaystyle \sum_{n \mathop = 0}^\infty \frac {\left({x}\right)^n} {n!} (n-1)!(-1)^{n-1}$

Rearranging by using the fact that

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

we get

$\displaystyle \sum_{n \mathop = 1}^\infty \frac {\left({-1}\right)^{n-1} } n x^n$

which is equivalent to

$\displaystyle \sum_{n \mathop = 1}^\infty \frac {\left({-1}\right)^{n+1} } n x^n$

This completes the proof

Also known as
The Newton-Mercator series is also known as the Mercator series.