Taylor series a. Use the definition of a Taylor series to find the first four...

snowlovelydayM

snowlovelydayM

Answered

2021-01-19

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

Answer & Explanation

pierretteA

pierretteA

Expert

2021-01-20Added 102 answers

Given :
The function is f(x)=2x and center at a=1
The definition of a Taylor series :
k=0ck(xa)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)=2x
f(x)=2xln(2)
f(x)=2xln2(2)
f(x)=2xln3(2)
f4(x)=2xln4(2)
To find derivative f(x) at x=1
f(1)=2
f(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=0ck(xa)k=k=02lnk(2)k!(x1)k
Jeffrey Jordon

Jeffrey Jordon

Expert

2021-12-17Added 2575 answers

Answer is given below (on video)

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