How do you find the Maclaurin series of the function #f(x)=5cos(10x^2)#?

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Answer 1 · 2015-06-06T15:58:56

The Maclaurin series of #f_{(x)}# is

#f_{(x)}=f(0)+f'(0)x+{f''(0)}/{2!} x^2+{f'''(0)}/{3!} x^3+ . . .#

Using this formula, get that the Maclaurin series of #cos(z)# is

#cos(z)=1-z^2/2+z^4/{4!}+....#

#5cos(z)=5-5/2z^2+5/{4!}z^4+....#

Substitute #z=10x^2#,

The Maclaurin series of #5cos(10x^2)# is then

#5cos(10x^2)=5-5/2(10x^2)^2+5/{4!}(10x^2)^4+ . . .#