Convergence of Infinite Continued Fractions
An infinite continued fraction is any continued fraction of the form
\[[ a_1,a_2,... ]\]
. For such a continued fraction, the Convergents of Continued Fractions \[C_n, \; n=1, \; 2, \; 3,...\]
tend to a limit.Proof
The first
\[n\]
convergents of the finite continued fraction \[[ a_1,a_2,... , \; a_n ]\]
are \[C_1, \; C_2,..., \; C_n \]
. Each convergent \[C_k = \frac{p_k}{q_k}\]
is defined by This Theorem.By the Relative Size of Convergents of Finite Continued Fractions of Convergents, the even convergents form a decreasing sequence, the odd convergents form an increasing sequence, and each even convergent
\[C_{2k}\]
is greater than the succeeding odd convergent \[C_{2k+1}\]
. The sequence of even convergents are bounded below by the odd convergents, and the sequence of odd convergents are bounded above by the even convergents, so each sequence converges to a limit.From the Properties of Convergents of Finite Continued Fractions, we get
\[C_{2k}-C_{2k-1}=\frac{1}{q_{2k}q_{2k-1}} \le \frac{1}{(2k)(2k-1)} \]
.since
\[q_k \ge k\]
for large \[k\]
.As
\[\]
the right hand side tends to zero, so the sequences tend to the same limit. 
