Great!

Well, yes, the condition for the geometric series expansion

to converge is that

holds.

Therefore, if

, we have that

, and thus the series

converges. But if

, we have

, thus

diverges.

By a purely algebraic transformation of that first term of the partial fraction decomposition we can use the (for

) convergent geometric series

, instead.

Simililarly, if we were told to develop the given function for

, we would have to transform the second term of the partial fraction decomposition in such a way that we can develop it in a, for

, convergent series of the form

.