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

1 Answer
George C.
Apr 15, 2017

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.

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