Question

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

Answer 1

Undefined or `0` - take your pick.

Explanation:

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.

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Footnote

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