The Taylor series for a function f about x=4 is given by n=1∑∞(n+1)3n(x−4)n+1=2⋅3(x−4)2+3⋅32(x−4)3+4⋅33(x−4)4+…+(n+1)3n(x−4)n+1+… and converges to f(x) on its interval of convergence.
6(A)Sequence Limits and Series Convergence TestsHard5 marks
Using the ratio test, find the interval of convergence of the Taylor series for f about x=4. Justify your answer.
6(B)Power Series, Radius of Convergence and Taylor SeriesEasy2 marks
Find the first three nonzero terms and the general term of the Taylor series for f′, the derivative of f, about x=4.
6(C)Power Series, Radius of Convergence and Taylor SeriesEasy1 mark
The Taylor series for f′ described in part B is a geometric series. For all x in the interval of convergence of the Taylor series for f′, show that f′(x)=7−xx−4.
6(D)Sequence Limits and Series Convergence TestsEasy1 mark
It is known that the radius of convergence of the Taylor series for f about x=4 is the same as the radius of convergence of the Taylor series for f′ about x=4. Does the Taylor series for f′ described in part B converge to f′(x)=7−xx−4 at x=8? Give a reason for your answer.