Use the definition of Taylor series to find the Taylor series, centered at c, for the function. f(x)=cos x, c=frac{-pi}{4}

York 2021-01-28 Answered
Use the definition of Taylor series to find the Taylor series, centered at c, for the function. f(x)=cosx, c=π4
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Expert Answer

grbavit
Answered 2021-01-29 Author has 109 answers
f(x)=cosx, c=π4
The Taylor polynomial for a function f(x) is
f(x)=f(c)+f(c)(xc)+f(c)2!(xc)2+f(c)3!(xc)3+...+f(n)cn!(xc)n
Now is
f(c)=cos(π4)
=12
f(c)=sin(π4)
=12
f(c)=cos(π4)
=12
f(c)=sin(π4)
=12
Then is
cosx=1212(xπ4)2(2!)(xπ4)2+2(3!)(xπ4)3+...+fn(c)(n!)(xπ4)n
=12+12(xπ4)+1(2)(2!)(xπ4)2+2(3!)(xπ4)3+...+fn(c)(n!)(xpi/4)n
=n=012(1)n(n+1)2n!(xπ4)n
Result
cosx=n=012(1)n(n+1)2n!(xπ4)n
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