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

Journal Article

### 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.

• 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

• http://ro.uow.edu.au/eispapers/5649

• 8

• 279

• 287

• 100

• 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.

• 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

• http://ro.uow.edu.au/eispapers/5649

• 8

• 279

• 287

• 100

• 548

### Place Of Publication

• United Kingdom