What is the radius of convergence of the MacLaurin series expansion for $f(x)= 1/sin x$ ?

Question

4348 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-04-15T20:57:15

Undefined or $0$ - take your pick.

The Maclaurin series for $f(x) = 1/sin(x)$ is undefined since $f(0)$ is undefined.

$lim_(x->0+) 1/sin(x) = +oo$

$lim_(x->0-) 1/sin(x) = -oo$

Even if we try to define $f(0)$ , then $f'(0)$ will be undefined.

So either the Maclaurin series is undefined or it will only describe $f(0)$ and have a zero radius of convergence.

$color(white)()$
Footnote

It would be possible to construct a Taylor series not centred at $npi$ with a positive radius of convergence.

What is the radius of convergence of the MacLaurin series expansion for f(x)= 1/sin xf(x)= 1/sin x?