## Periodic Infinite Continued Fractions

An infinite continued is fraction is of the formThe continued fraction always converges. In fact the numerators p-k of the kth convergents forms a decreasing sequence and the denominators form an increasing sequence, so the quotient p-k over q-k converges to a limit. Given a periodic infinite continued fraction, we can find the value of the fraction as in the following example.

In this continued fraction the sequence 313 generates the fraction so we can write

Hence

We can rearrange this into a quadratic equation.

Clearing the fractions now giveswhich simplifies towhich has solutionsis positive so we ignore the negative square root which gives a negative value forto obtain

Example: Find the value of

LetthenThis has solutionstaking the positive square root. Then