Is there any simpler way of writing #1/(x+1/((x+1)+1/((x+2)+1/((x+3)+ddots#, where #x=RR#?
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, . . . ]#
See below.
Explanation:
Calling