One of the immediate applications of the convergents is that they give rational approximations given the continued fraction of a number. This allows finding rational approximations to irrational numbers.
Convergents of continued fractions can be calculated in sage
sage: cf = continued_fraction(17/11)
sage: convergents = cf.convergents()
Continued fractions have many other applications. One such applicable in cryptology is based on Legendre's theorem in diophantine approximations.
is a convergent of
Wiener's attack on the RSA cryptosystem works by proving that under certain conditions, an equation of the form
could be derived where
is entirely made up of public information and
is made up of private information. Under assumed conditions, the inequality
is statisfied, and the value
(private information) is calculated from convergents of
(public information), consequently breaking the RSA cryptosystem.