Newton-Mercator Series

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Theorem

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

Then:

\(\displaystyle \ln \left({1 + x}\right)\) \(=\) \(\displaystyle x - \dfrac {x^2} 2 + \dfrac {x^3} 3 - \dfrac {x^4} 4 + \cdots\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{n \mathop = 1}^\infty \frac {\left({-1}\right)^{n + 1} } n x^n\)

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


This is known as the Newton-Mercator series.


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 $\left\vert{x}\right\vert <1$

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:

$\ln(x+1)=\displaystyle \int_0^x \frac {\mathrm dt} {1+t}$


Combining these facts, we get:

$\ln(x+1)=\displaystyle \int_0^x \displaystyle \sum_{n \mathop = 0}^\infty (-1)^n t^n dt$


From Linear Combination of Integrals, we can rearrange this to

$\displaystyle \sum_{n \mathop = 0}^\infty (-1)^n \displaystyle \int_0^x t^n dt$

Then, using Integral of Power:

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

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

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

This is equivalent to:

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


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


For $x=-1$, we get:

$\displaystyle \sum_{n \mathop = 1}^\infty \dfrac {(-1)^{n+1}} {n} (-1)^n$


$(-1)^{n+1}$ and $(-1)^n$ 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 {(-1)^{n+1}} {n} $


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


Therefore, we can conclude that:

$\ln(x+1)=\displaystyle \sum_{n \mathop = 1}^\infty \dfrac {(-1)^{n+1}} {n} x^{n}$ for $-1 < x \le 1$.

$\blacksquare$


Examples

Newton-Mercator Series: $\ln 2$

$\ln 2 = 1 - \dfrac 1 2 + \dfrac 1 3 - \dfrac 1 4 + \dfrac 1 5 - \cdots$


Also known as

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


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.

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