The Ratio Test can be used to find out, for what values of x a given power series
converges. It works by comparing the given power series to the geometric series. Recall that the geometric series
is convergent exactly when -1<q<1.
Let's consider a series (no power yet!) and be patient for a couple of moments:
Suppose that all s are positive and that there is a q<1 so that
Then we know that ; we also know that ; in the next step we can recursively conclude that ; in general we obtain
Thus we can conclude that the series under consideration converges:
The partial sums are all caught between the leftmost part of the inequality (0) and the rightmost part . Since the partial sums are increasing, the series has no choice but to converge!
Maybe a picture helps. The series under consideration is depicted in blue; it is caught between 0 (in red) and the geometric series (in black), which itself is bounded by its limit (in red).
Click here for the answer.