Chapter 11.3, Problem 67E

a.

To determine

Whether the statement “The function $f\left(x\right)=\sqrt{x}$ has a Taylor series centered at 0 is true or false.

b.

To determine

Whether the statement “The function $f\left(x\right)=\csc x$ has a Taylor series centered at $\frac{\pi }{2}$” is true or false.

c.

To determine

Whether the statement “If f has a Taylor series that converges only on $\left(-2,2\right)$, then $f\left(x^2\right)$ has a Taylor series that also converges only on $\left(-2,2\right)$” is true or false.

d.

To determine

Whether the statement “If $p\left(x\right)$ is the Taylor series for f centered at 0 then $p\left(x-1\right)$ is the Taylor series for f centered at 1” is true or false.

e.

To determine

Whether the statement “The Taylor series for an even function about 0 has only even powers of x” is true or false.