The Radius of Convergence

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 tex2html_wrap_inline64 . 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).

Try it yourself!

Why are the partial sums increasing? Hint: Read this page again carefully!

Click here for the answer.

Helmut Knaust
Wed Jul 10 22:47:37 MDT 1996