Relative Size of Convergents of Finite Continued Fractions
For any finite continued fraction
\[[ a_1,a_2,...,a_{n-1},a_n ]\]
, the odd convergents \[C_{2r-1}\]
form a strictly increasing sequence \[C_1 \lt C_3 \lt C_5 ...\]
and the even convergents \[C_{2r}\]
form a strictly decreasing sequence \[C_2 \gt C_4 \gt C_6 ...\]
/Br /> Also every even convergent is greater than every odd convergent.Proof
From the Properties of Convergents of Finite Continued Fractions,
\[a_k \gt 0\]
for for \[k \ge 2\]
and \[q_k \gt 0\]
for \[k \ge 1\]
, so for \[k \ge 3\]
, \[C_k-C_{k-2}=\frac{p_k}{q_k}-\frac{p_{k-2}}{q_{k-2}}=\frac{(-1)^{k-1}a_k}{q_kq_{k-1}}\]
has the same sign as \[(-1)^{k-1}\]
so is positive for odd \[k\]
and negative for even \[k\]
, showing that the odd convergents strictly increase and the even convergents strictly decrease.\[C_k-C_{k-1}=\frac{p_k}{q_k}-\frac{p_{k-1}}{q_{k-1}}=\frac{(-1)^{k}}{q_kq_{k-1}}\]
Hence the even convergent is larger than the succeeding odd convergent.
