2

Let's say I have a continued fraction $a = [a_1, a_2, \dots, a_n]$ but I make a mistake and switch the digit at two places, do I get a number which is bigger or smaller?

For $a,b \in \mathbb{N}$ which continued fraction is bigger $[1,1,a,1,1,1,1]$ or $[1,1,1,b,1,1,1]$ ?

See: Continued Fraction [1,1,1,...]

Community
  • 1
cactus314
  • 24,438
  • 2
    Increasing the value in an odd-numbered place (starting the counting at $1$) results in larger fraction, increasing the value in an even-numbered place results in smaller fraction. Changes made to lower number (early) spots outweighs any change to any higher-number spot. – Arthur Jun 07 '14 at 19:33

1 Answers1

1

Hints:

  1. $a<[a,{\cal S}]<a+1$ for any natural $a$ and nonempty sequence $\cal S$ of naturals after.
  2. $[{\cal S}_1]>[{\cal S}_2]\implies [u,{\cal S}_1]<[v,{\cal S}_2]$ for any $u\le v$ and sequences ${\cal S}_{1,2}$ of naturals.

In fact, to determine which of two (each possibly finite, possibly infinite) continued fractions is bigger, it suffices to look for the first number in which they differ. In an odd position, the larger digit is attached to the larger number, and in an even position the opposite holds. Another way of saying this is with Arthur's comment above. This can also be proved with the above two hints.

anon
  • 151,657