A harmonic expansion of a real number series
(Havil 2003, p. 99), where floor function .
The coefficients fractional
part of floor
function at each step, and continuing with the new fractional
part .
Examples are given in the following table.
OEIS harmonic expansion A054977 [2, 1, 1, 1, 1, 1, 1, ...] A096622 [0, 1, 0, 1, 4, 1, 4, ...] A075874 [3, 0, 0, 3, 1, 5, 6, 5, ...]
See also Continued Fraction ,
Engel
Expansion ,
Harmonic Series
Explore with Wolfram|Alpha
References Havil, J. "Harmonic Expansion." §11.8 in Gamma:
Exploring Euler's Constant. Sloane, N. J. A. Sequences A054977 ,
A075874 , and A096622
in "The On-Line Encyclopedia of Integer Sequences." Referenced
on Wolfram|Alpha Harmonic Expansion
Cite this as:
Weisstein, Eric W. "Harmonic Expansion."
From MathWorld https://mathworld.wolfram.com/HarmonicExpansion.html
Subject classifications
is a series ![[n;a,b,c,...]](/images/equations/HarmonicExpansion/Inline2.svg)
defined by
and 
is the floor function.
The coefficients 
, 
, 
, ... are obtained from the fractional
part of 
by successively multiplying by 2, 3, 4, ..., taking the floor
function at each step, and continuing with the new fractional
part.
