# How do you find the maclaurin series expansion of f(x)=cos3x?

Refer to explanation

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A Maclaurin series is a Taylor series expansion of a function about 0,
hence

##f(x)=f(0)+f'(0)x+(f”(0))/(2!)*x^2+(f”'(0))/(3!)+…+(f^(n)(0))/(n!)*x^n##

where ##f^(n)(0)## is the n-th order derivative of ##f(x)##.

Hence we have to calculate some derivatives around zero so

##f(x)=cos3x=>f(0)=1##
##f'(x)=-3sin3x=>f'(0)=0##
##f”(x)=-3^2*cos3x=>f”(0)=-3^2##
##f”'(x)=3^3*sin3x=>f”'(0)=0##
##f””(4)=3^4*cos3x=>f””(0)=3^4##

So the maclaurin series can be written as

##cos3x=Σ_0^oo (-1)^n*(3)^(2n)*(x^(2n))/((2n)!)##

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