Umformung von Termen: 1/(1+(1/(1+ 1/(1+x)))

Question

Wie kann ich diesen Term umformen?

$$ \frac{1}{1+\frac{1}{1+\frac{1}{1+x}}}= \cdots $$

Answer 1

Hi, man könnte den Kettenbruch schrittweise auflösen:
$$ \frac { 1 } { 1 + \frac { 1 } { 1 + \frac { 1 } { 1+x } } } =  \frac { 1 } { 1 + \frac { 1 } { \frac { 2+x } { 1+x } } } =  \frac { 1 } { 1 + \frac { 1+x } { 2+x } } =  \frac { 1 } { \frac { 3+2x } { 2+x } } =  \frac { 2+x } { 3+2x }. $$

Answer 2

Beginne unten mit Bruchadditionen:

1/(1+(1/(1+ 1/(1+x)))

= 1/(1+(1/((1+x)/(1+x))+ 1/(1+x)))

= 1/(1+(1/((1+x+ 1)/(1+x)))

= 1/(1+(1/((x+ 2)/(1+x)))

= 1/(1+(x+ 1)/(x+2))

= 1/((x+2)/(x+2) +(x+ 1)/(x+2))

= 1/((x+2) +(x+ 1)/(x+2))

= 1/((2x+3)/(x+2))

= (x+2)/(2x+3)

Kontrolle:

https://www.wolframalpha.com/input/?i=1%2F%281%2B%281%2F%281%2B+1%2F%281%2Bx%29%29%29