Show the convergence of the infinite simple continued fraction in Problem 2, representing the recurrence relation .
Let be the fraction represented by the th term in the continued fraction. Thus are the even convergents; are the odd convergents; both and satisfy the recurrence relation.
One needs to show that is bounded and monotonic increasing and is bounded and monotonic decreasing, and that these two subsequences converge to the same limit.
A formula for the difference of convergents will help:
Expanding the numerator on the right side by substitution from the recurrence relation,
Repeat the substitution on the right until the result is in terms of the sequence initial constants:
Since , and is monotonic increasing.
Similarly and is monotonic decreasing.
Both subsequences are bounded, for and .
Thus it remains to show that both subsequences converge to the same limit, by showing the difference between the sequences tends to for large . Observing that when :