What are the first 3 nonzero terms in the Taylor series expansion about x = 0 for the function #f(x)=cos(4x)#?

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Answer 1 · 2015-03-17T21:15:08

Answer. #1 - 8 x^2 + 32/3 x^4#.

Because

#cos(X) = 1 - X^2/2 + X^4/24 + o_{X->0}(X^4)#,

and because #4x ->0# when #x->0#, you get

#cos(4x) = 1 - (4x)^2/2 + (4x)^4/24 + o_{x->0}((4x)^4)#,

so

#cos(4x) = 1 - 8 x^2 + 32/3 x^4 + o_{x->0}(x^4)#,