It exists a general Taylor expansion which is known as 'Binomial series' ...
(1)
... where...
(2)
Setting in (1) we obtain...
(3)
Kind regards
Some examples of 'remainder'...
http://mathworld.wolfram.com/LagrangeRemainder.html
http://mathworld.wolfram.com/CauchyRemainder.html
http://mathworld.wolfram.com/SchloemilchRemainder.html
... but they are not 'alone'... what is your professor's 'preference'?...
Kind regards