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 tex2html_wrap_inline52 s are positive and that there is a q<1 so that


Then we know that tex2html_wrap_inline56 ; we also know that tex2html_wrap_inline58 ; in the next step we can recursively conclude that tex2html_wrap_inline60 ; in general we obtain


Thus we can conclude that the series under consideration converges:


The partial sums tex2html_wrap_inline62 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

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