Find a power series representation for the subsequent function and find out its interval of convergence.

g (x) = 1/1+x³

Solution

What we require to do here is to relate this function back to f (x) = 1/ (1-x). In fact this is easier than it might look. Remind that the x in f (x) = 1/ (1-x) is simply a variable and can represent anything. Thus, a quick rewrite of g(x) gives,

g (x) = 1/ 1- (-x³)

and so the - x³ in  g (x) holds the same place as the x in f (x) = 1/ (1-x). Therefore, all we need to do is replace the x in

and we've got a power series representation for g (x).

Note: we replaced both the x in the power series and in the interval of convergence. 

All we need to do now is a modest simplification.

Thus, in this case the interval of convergence is similar as the original power series.  This generally won't happen. Much more frequently, than not the new interval of convergence will be dissimilar from the original interval of convergence.