Subject: Power series representations

Is there a systematic way of finding a power series representation of a function? I understand that you have to manipulate the function so that it is of the form 1/(1-x), but beyond that I am lost.

Grace
secondary question,
student

There is a systematic way of finding a power series representation of some functions that is usually learned in a second university calculus class. If, however, you can manipulate the function so that it is in the form 1/(1-x) the procedure is more straightforward.

If you divide 1-x into 1 using long division you see a power series representation

which is the geometric series wirh first term 1 and the common ratio x.

This is not the end of the story as the series converges only if |x| < 1. An explanation of why this is true can be found in Infinite Geometric Series, the answer to a previous question on series.