Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.b. The function f(x) = csc x has a Taylor series centered at π/2.

Accepted Answer

Recall that a Taylor series of a function $$f(x)$$ centered at $$a is $$given by the infinite sum $$\sum_{n=0}^{\infty} \frac{f$$^{(n)}$$(a)}{n!} (x - a)^n$$, where $$f^{(n)}(a$$)$$ denotes the n$-th derivative of f$ evaluated at x = a$. To determine if f(x$$) = \csc x$$ has a Taylor series centered at x$$ = \frac{\pi}{2}$$, first check if f$ is differentiable at x$$ = \frac{\pi}{2}$$ and in some open interval around it. Evaluate f$$\left(\frac{\pi}{2}ight) = \csc\left(\frac{\pi}{2}ight) = 1$$, which is finite, so the function is defined at x$$ = \frac{\pi}{2}$$. Next, consider the behavior of $$\csc x$$ near x$$ = \frac{\pi}{2}$$. Since $$\sin x$$ is nonzero and smooth around $$\frac{\pi}{2}$$, $$\csc x = \frac{1}{\sin x}$$ is also smooth and infinitely differentiable there. Therefore, f(x$$) = \csc x$$ is infinitely differentiable at x$$ = \frac{\pi}{2}$$, so it does have a Taylor series centered at $$\frac{\pi}{2}$$. The Taylor series converges to f(x$$)$$ in some interval around $$\frac{\pi}{2}$$ where $$\sin x 
eq 0$$.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. The function f(x) = csc x has a Taylor series centered at π/2.

Verified video answer for a similar problem:

Key Concepts

Taylor Series and Center of Expansion

Domain and Differentiability of csc(x)

Radius of Convergence and Singularities