# Is there any simpler way of writing #1/(x+1/((x+1)+1/((x+2)+1/((x+3)+ddots#, where #x=RR#?

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For example, #1/(1+1/(2+1/(3+1/(4+ddots#

For example,

##### 2 Answers

In terms of notation, sure. If you wanted to denote an infinite continued fraction, i.e.

#x + 1/(x+1 + 1/(x+2 + 1/(x + 3 + 1/(x + 4 + ddots)))# ,

you would write

#[x; x + 1, x + 2, x + 3, . . . ]#

For example, if you represent

#15 + 20/27# ,

or by multiplying through by

#(425)/27 = color(red)(15) + 1/(color(red)(1) + 1/(color(red)(2) + 1/(color(red)(1) + 1/color(red)(6)))# ,

or

#[15; 1, 2, 1, 6]#

So, for your example, to write

#1/(1 + 1/(2 + 1/(3 + 1/(4 + ddots))))# ,

you could write

#[0; 1, 2, 3, 4, . . . ]#

#### Answer:

See below.

#### Explanation:

Calling