Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Rn(x)→0 f(x)=2cos⁡(x),a=3π

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Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion.  Rn(x)→0  f(x)=2cos⁡(x),a=3π

Alix Ortiz · Accepted Answer

It is given thatf(x)=2cos⁡(x),a=3πThe general form for a Taylor series isf(x)=∑n=0∞fn(a)n!(x−a)n=f(a)+f′(a)1!(x−a)+f″(a)2!(x−a)2+f‴(a)3!(x−a)3+...Obtain the table of values of x and nth power of f at the given point a as shown below:nfnfn(3π)02cos⁡x2cos⁡(3π)=2(−1)=−21−2sin⁡x−2sin⁡x=−2(0)=02−2cos⁡x−2cos⁡(3π)=−2(−1)=232sin⁡x2sin⁡x=2(0)=042cos⁡x2cos⁡(3π)=2(−1)=−25−2sin⁡x−2sin⁡x=−2(0)=0Using the definition of Taylor series, obtain the Taylor series for the given function as shown below:f(x)=∑n=0∞fn(a)n!(x−a)n=f(3π)+f′(3π)1!(x−3π)+f″(3π)2!(x−3π)2+f‴(3π)3!(x−3π)3+...=−2+0+2(x−3π)2+0(−2)(x−3π)4+0+(2)(x−3π)6=∑n=0∞(−2)n+1(x−3π)2n+2

Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. R_n(x)to0 f(x)=2cos(x),a=3pi

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