Question #5d213

1 Answer
Sasha P.
Sep 28, 2015

See the explanation.

Explanation:

#f(x)=sum_(k=0)^n f^((k))(0) x^k/(k!) + f^((n+1))(epsilon)x^(n+1)/((n+1)!)#

#f(x)=cos^2x#

#f(0)=1#
#f'(x)=-2cosxsinx=-sin2x => f'(0)=0#
#f''(x)=-2cos2x => f''(0)=-2#
#f^((3))(x)=4sin2x => f^((3))(0)=0#
#f^((4))(x)=8cos2x => f^((4))(0)=8#
#f^((5))(x)=-16sin2x => f^((5))(0)=0#
#f^((6))(x)=-32cos2x => f^((6))(0)=-32#

and so on...

#f(x)=f(0)+f'(0) x^1/(1!)+f''(0) x^2/(2!)+f^((3))(0) x^3/(3!)+...#

#f(x)=1+0-2x^2/(2!)+0+8x^4/(4!)+0-32x^6/(6!)+....#
#f(x)=1+0-2^1x^2/(2!)+0+2^3x^4/(4!)+0-2^5x^6/(6!)+....#
#f(x)=1+0-2^(2-1)x^2/(2!)+0+2^(4-1)x^4/(4!)+0-2^(6-1)x^6/(6!)+....#

#f(x)=1+sum_(k=1)^oo (-1)^k 2^(2k-1) x^(2k)/((2k)!)#

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