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A simple proof of Euler's continued fraction of e^{1/M}

Journal Article


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Abstract


  • A continued fraction is an expression of the form

    f0+ g0

    f1+g1

    f2+g2

    and we will denote it by the notation [f0, (g0, f1), (g1, f2), (g2, f3), … ]. If the numerators g i 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 g i coefficients equal to 1 and with all the f i 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

    [f0, (g0,f1), (g1,f2), (g2,f3),...] = lim [f0, (g0,f1), (g1,f2),..., (gk-1,fk)]

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

Publication Date


  • 2016

Citation


  • Tonien, J. (2016). A simple proof of Euler's continued fraction of e^{1/M}. The Mathematical Gazette, 100 (548), 279-287.

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=6677&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/5649

Number Of Pages


  • 8

Start Page


  • 279

End Page


  • 287

Volume


  • 100

Issue


  • 548

Place Of Publication


  • United Kingdom

Abstract


  • A continued fraction is an expression of the form

    f0+ g0

    f1+g1

    f2+g2

    and we will denote it by the notation [f0, (g0, f1), (g1, f2), (g2, f3), … ]. If the numerators g i 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 g i coefficients equal to 1 and with all the f i 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

    [f0, (g0,f1), (g1,f2), (g2,f3),...] = lim [f0, (g0,f1), (g1,f2),..., (gk-1,fk)]

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

Publication Date


  • 2016

Citation


  • Tonien, J. (2016). A simple proof of Euler's continued fraction of e^{1/M}. The Mathematical Gazette, 100 (548), 279-287.

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=6677&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/5649

Number Of Pages


  • 8

Start Page


  • 279

End Page


  • 287

Volume


  • 100

Issue


  • 548

Place Of Publication


  • United Kingdom