40A05-HalleysFormula.tex

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\pmtitle{Halley's formula}
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\begin{document}
The following formula is due to the English scientist and mathematician Edmond Halley (1656 \`a 1742):
\begin{align}
\ln{x} \;=\; \lim_{n\to\infty}(\sqrt[n]{x}-1)n
\end{align}

{\it Proof.}\, We change the $n$th root to power of $e$ and use the power series expansion of exponential function:
\begin{align*}
(\sqrt[n]{x}-1)n &\;=\; (e^{\frac{\ln{x}}{n}}-1)n\\
&\;=\; \left(\sum_{m=0}^\infty\frac{(\ln{x}/n)^m)}{m!}-1\right)\!n\\ 
&\;=\; \sum_{m=1}^\infty\frac{(\ln{x}/n)^mn}{m!}\\
&\;=\; \ln{x}+\frac{1}{n}\sum_{m=2}^\infty\frac{(\ln{x})^m}{m!n^{m-2}}
\end{align*}
The last converging series has a finite sum, and as\, $n \to \infty$,\, the asserted formula follows.\\

\textbf{Note.}\, The formula (1) was known also by Leonhard Euler, who used it for defining the natural logarithm.\, Using  (1), one can easily prove the well-known laws of logarithm, e.g.
\begin{align*}
\ln{xy} &\;=\; \lim_{n\to\infty}(\sqrt[n]{x}\sqrt[n]{y}-1)n\\
&\;=\; \lim_{n\to\infty}(\sqrt[n]{x}\sqrt[n]{y}-\sqrt[n]{y}+\sqrt[n]{y}-1)n\\
&\;=\; \lim_{n\to\infty}y^{\frac{1}{n}}(\sqrt[n]{x}-1)n+\lim_{n\to\infty}(\sqrt[n]{y}-1)n\\
&\;=\; y^0\ln{x}+\ln{y}\\
&\;=\; \ln{x}+\ln{y}.
\end{align*}


\begin{thebibliography}{8}
\bibitem{L}{\sc Paul Loya}: {\em Amazing and Aesthetic Aspect of Analysis: On the incredible infinite}. A course in undergraduate analysis, fall 2006.  Available \PMlinkexternal{here}{http://www.math.binghamton.edu/dennis/478.f07/EleAna.pdf}.
\end{thebibliography}

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