That is a classical expansion by Euler (sadly not even I know the proof).

But I hope you are asking not to prove it but to get 4 decimals?

Using a well-known inequlality from continued fractions:

|x-p_n/q_n|<=(1/q_n)^2

In order to get 4 decimal points we require that,

(1/q_n)^2<=.0001

Equivalently,

1/q_n<=.0316

Thus,

q_n>=31.62

Thus,

q_n>=32

That is the smallest "n" that makes the denominator of the convergent at least 32.