Finding MacLaurin series

Question

Find the MacLauren series for the function f(x)=sin(x)-x+(x^3)/3!

Answer 1

f⁽⁰⁾(x)=sin(x)-x+x³/3!; f⁽⁰⁾(0)=0

f⁽¹⁾(x)=cos(x)-1+x²/2!; f⁽¹⁾(0)=0

f⁽²⁾(x)=-sin(x)+x; f⁽²⁾(0)=0

f⁽³⁾(x)=-cos(x)+1; f⁽³⁾(0)=0

f⁽⁴⁾(x)=sin(x); f⁽⁴⁾(0)=0

f⁽⁵⁾(x)=cos(x); f⁽⁵⁾(0)=1

f⁽⁶⁾(x)=-sin(x); f⁽⁶⁾(0)=0

f⁽⁷⁾(x)=-cos(x); f⁽⁷⁾(0)=-1

f⁽⁸⁾(x)=f⁽⁴⁾(x), f⁽⁹⁾(x)=f⁽⁵⁾(x), etc.

Series is x⁵/5!-x⁷/7!+x⁹/9!-...+(-1)ⁿx²ⁿ⁺⁵/(2n+5)! = ∑[n∈[0,∞)](-1)ⁿx²ⁿ⁺⁵/(2n+5)!