Definition:Newton-Mercator Series

Definition

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

Then:

\(\ds \map \ln {1 + x}\)

\(=\)

\(\ds x - \dfrac {x^2} 2 + \dfrac {x^3} 3 - \dfrac {x^4} 4 + \cdots\)

\(\ds \)

\(=\)

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

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

This is known as the Newton-Mercator series.

Also known as

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

Some sources call it the logarithmic series.

Examples

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.

Also see

• Power Series Expansion for $\map \ln {1 + x}$ for a proof of its convergence

• Results about the Newton-Mercator series can be found here.

Source of Name

This entry was named for Isaac Newton and Nicholas Mercator.

Historical Note

The Newton-Mercator series was discovered independently by both Isaac Newton and Nicholas Mercator, the latter of whom published it in $1668$.

However, it was also independently discovered by Grégoire de Saint-Vincent.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): logarithmic series
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Mercator's series
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): logarithmic series
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Mercator's series