Relative Size of Convergents of Finite Continued Fractions

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

Hence the even convergent is larger than the succeeding odd convergent.

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