i understand that the number pattern is 1, -2, 8, -32, etc with values only in the even derivatives and 0.

However, the actual problem for me is to actually write the series. like i know it would look something like (-1)^n (??x^(2n))/(2n!)

The ?? I'm having a hard time with since that sequence of numbers (1, -2, 8...) i can't figure out. Thank you