Newton-Mercator Series/Examples/2

Theorem
The Newton-Mercator Series for $x = 1$ converges to the natural logarithm of $2$:

This real number is known as Mercator's constant.

Proof
From the definition of the Newton-Mercator Series:

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

Setting $x = 1$:

The fact that it does indeed converge is shown in Alternating Harmonic Series is Conditionally Convergent.