Why the value of this kind of sequence always get bigger and then smaller

Question

I tried to use $1$, $2$ and $3$ into this formula, and for all the answer will get bigger first than smaller. I want to know the reason for it.

Answer 1

These are simple continued fractions. This from wikipedia tells you the approximation errors alternate in sign:

Corollary 1: The even convergents continually increase, but are always less than $x$.

Corollary 2: The odd convergents continually decrease, but are always greater than $x$ .

(Good for you for noticing this.)

https://en.wikipedia.org/wiki/Continued_fraction#Properties