Write the first five terms of the sequence.
a(n) = ((-1)^n x^(2n + 1))/(2n + 1)!
Not sure about how to deal with the x here. Will it just remain in the sequence?
Sure; it's just part of the expression. So, assuming we start with n=1, the powers and denominators go up as
3,5,7,9,...
So, the sequence is
-x^3/3!, x^5/5!, -x^6/7!, ...
though I suspect that maybe we start at n=0. But you can fix it if that's the case.