There is a formula, which I suspect you were expected to know, that says the error, in cutting off a Maclaurin series of a function at the nth power, is less than or equal to where "M" is an upper bound on the n+1 derivative of the function between 0 and x.

Here, since every derivative is again and that has a maximum of 1 on any interval [-x, 0], that maximum error is . That is the same as saying that
8!= 40320< 1000000 and 9!= 362880> 1000000 so you need 9 terms.