Newton-Mercator Series/Examples

Examples of the Newton-Mercator Series

Newton-Mercator Series: $\ln 2$

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

\(\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^\paren {n - 1} } n\)

\(=\)

\(\ds 1 - \frac 1 2 + \frac 1 3 - \frac 1 4 + \dotsb\)

\(\ds \)

\(=\)

\(\ds \ln 2\)

This real number is known as Mercator's constant.