Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.a. The function has a Taylor series centered at 0.b. The function f(x) = csc x has a Taylor series centered at ?/2.c. If f has a Taylor series that converges only on (?2, 2), then f(x2)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.

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 .Step 2:(b).The function has a Taylor series centered at /2.The general equation of a taylor series centered at is Here This continues.Thus the taylor expansion of Thus the given statement is true.Step 3:(c). . If f has a Taylor series that converges only on (2, 2), then has a Taylor series that also converges only on (2, 2).f has a Taylor series that converges only on (2, 2), implies that is the radius of convergence.Therefore the radius of convergence of will be Thus has a Taylor series that converge on .