Square root convergents

It is possible to show that the square root of two can be expressed as an infinite continued fraction.

\[ \sqrt 2 =1+ \frac 1 {2+ \frac 1 {2 +\frac 1 {2+ \dots}}} \]

By expanding this for the first four iterations, we get:

\( 1 + \frac 1 2 = \frac 3 2 = 1.5 \)

\( 1 + \frac 1 {2 + \frac 1 2} = \frac 7 5 = 1.4 \)

\( 1 + \frac 1 {2 + \frac 1 {2+\frac 1 2}} = \frac {17}{12} = 1.41666 \dots \)

\( 1 + \frac 1 {2 + \frac 1 {2+\frac 1 {2+\frac 1 2}}} = \frac {41}{29} = 1.41379 \dots \)

The next three expansions are \( \frac{99}{70} \), \( \frac{239}{169} \), and \( \frac{577}{408} \), but the eighth expansion, \( \frac{1393}{985} \), is the first example where the number of digits in the numerator exceeds the number of digits in the denominator.

In the first one-thousand expansions, how many fractions contain a numerator with more digits than the denominator?