I'm in an analysis class currently discussing differentiation. I'm pretty sure I need to use Taylor's Theorem on these but I'm not sure. Could someone work me through them?
First, if x is greater than zero, show that (1+x)^(1/3)-(1+(1/3)x-(1/9)x^2) is less than or equal to (5/81)x^3.
Second, if abs(x) is less than or equal to one, show that abs([sin(x)-(x-(x^3)/6+(x^5)/120]) is less than (1/5040).
Sorry. I know it's quite complicated, but I have no idea. Anyways, thank you for any help you can give.