Is there a function that, given a rational function, will return the general term of its infinite series expansion?

Question

Is here some way to expand a rational function to an infinite sum in Mathematica, i.e., a series? I want the general term of the series.

For example, $\dfrac{2}{3(x-1)^3}$

Series works, but I was wondering if there was a function that converts the fraction straight to an infinite sum, in sum notation, not expanded out. — Pablo, Aug 03 '14 at 13:18
@Henry: do you mean "rational function" (or even just "function") rather than "fraction"? — murray, Aug 03 '14 at 14:44

score 11 · Answer 1 · answered Aug 03 '14 at 13:53

In Mathematica 10 you can use SeriesCoefficient and Inactive to get what you require

Inactive[Sum][SeriesCoefficient[2/(3(x-1)^3),{x,0,n},Assumptions->n>=0]x^n,{n,0,\[Infinity]}]

where Inactive prevents Sum from evaluating.

You can then "activate" the Sum as follows

Activate[%]

to get back to your original expression.

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