The Taylor expansion of the 'gaussian function' is...

(1)

... and integrating it 'term by term'...

(2)

The (2) is an 'alternating terms' series, so that the error can't be greater than the first term you neglect. The condition for error then becomes...

(3)

... and that happens just for so that the first 4 tems of (2) will give the requested accuracy...

Kind regards