Taylor series a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. f(x)=2x,a=1

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Taylor series  a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.  b. Write the power series using summation notation.  f(x)=2x,a=1

pierretteA · Accepted Answer

Given :The function is f(x)=2x and center at a=1The definition of a Taylor series :∑k=0∞ck(x−a)k, where ck=fk(a)k!, for k=0,1,2,3,...a)To use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a Find derivative of the function .f(x)=2xf′(x)=2xln⁡(2)f″(x)=2xln2⁡(2)f‴(x)=2xln3⁡(2)f4(x)=2xln4⁡(2)To find derivative f(x) at x=1f(1)=2f′(1)=2ln⁡(2)f″(1)=2ln2⁡(2)f‴(1)=2ln3⁡(2)f4(1)=2ln4⁡(2)To find the first four nonzero terms of the Taylor series .c0=f(1)=2,c1=f′(1)1!=2ln⁡(2)1!,c2=f″(1)2!=2ln2⁡(2)2!,c3=f‴(1)3!=2ln3⁡(2)3!,c4=f4(1)4!=2ln4⁡(2)4!The in general ck=2lnk⁡(2)k!b) To write the power series using summation notation .By using definition of a Taylor series and part a.The series for centered at  a=1 is∑k=0∞ck(x−a)k=∑k=0∞2lnk⁡(2)k!(x−1)k

Jeffrey Jordon · Accepted Answer

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Taylor series a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. f(x)=2^x,a=1

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