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A simple proof of Euler's continued fraction of e1/M

Published online by Cambridge University Press:  14 June 2016

Joseph Tonien
Joseph Tonien*
Affiliation:
School of Computing and Information Technology, University of Wollongong, Australia e-mail: [email protected]
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Extract

A continued fraction is an expression of the form

and we will denote it by the notation [f0, (g0, f1), (g1, f2), (g2, f3), … ]. If the numerators gi are all equal to 1 then we will use a shorter notation [f0, f1, f2, f3, … ]. A simple continued fraction is a continued fraction with all the gi coefficients equal to 1 and with all the fi coefficients positive integers except perhaps f0.

The finite continued fraction [f0, (g0, f1), (g1, f2),…, (gk–1, fk)] is called the k th convergent of the infinite continued fraction [f0, (g0, f1), (g1, f2),…]. We define

if this limit exists and in this case we say that the infinite continued fraction converges.

Type
Research Article
Information
The Mathematical Gazette , Volume 100 , Issue 548 , July 2016 , pp. 279 - 287
Copyright
Copyright © Mathematical Association 2016 

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References

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