Halley’s formula

The following formula is due to the English scientist and mathematician Edmond Halley (1656 à 1742):

lnx=limn(xn-1)n (1)

Proof.  We change the nth root to power of e and use the power seriesMathworldPlanetmath expansion of exponential functionDlmfDlmfMathworldPlanetmathPlanetmath:

(xn-1)n =(elnxn-1)n

The last converging series has a finite sum, and as  n,  the asserted formula follows.

Note.  The formula (1) was known also by Leonhard Euler, who used it for defining the natural logarithmMathworldPlanetmathPlanetmathPlanetmath.  Using (1), one can easily prove the well-known laws of logarithm, e.g.

lnxy =limn(xnyn-1)n


  • 1 Paul Loya: Amazing and Aesthetic Aspect of Analysis: On the incredible infinite. A course in undergraduate analysis, fall 2006. Available http://www.math.binghamton.edu/dennis/478.f07/EleAna.pdfhere.
Title Halley’s formula
Canonical name HalleysFormula
Date of creation 2013-03-22 19:34:39
Last modified on 2013-03-22 19:34:39
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Result
Classification msc 40A05
Related topic ListOfCommonLimits