Solution Found!
Explain why or why not Determine whether the
Chapter 8, Problem 51E(choose chapter or problem)
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The function \(f(x)=\sqrt{x}\) has a Taylor series centered at 0.
b. The function f(x) = csc x has a Taylor series centered at \(\pi / 2\).
c. If f has a Taylor series that converges only on (-2,2), then \(f\left(x^{2}\right)\) has a Taylor series that also converges only on (-2,2).
d. If p is the Taylor series for f centered at 0, then p(x - 1) is the Taylor series for f centered at 1.
e. The Taylor series for an even function about 0 has only even powers of x.
Questions & Answers
QUESTION:
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The function \(f(x)=\sqrt{x}\) has a Taylor series centered at 0.
b. The function f(x) = csc x has a Taylor series centered at \(\pi / 2\).
c. If f has a Taylor series that converges only on (-2,2), then \(f\left(x^{2}\right)\) has a Taylor series that also converges only on (-2,2).
d. If p is the Taylor series for f centered at 0, then p(x - 1) is the Taylor series for f centered at 1.
e. The Taylor series for an even function about 0 has only even powers of x.
ANSWER:Solution 51EStep 1:(a). The function has a Taylor series centered at 0.The general equation of a taylor series centered at 0 is Here and the differentiation of at 0 is always 0.Thus this is a false statement .