The Radius of Convergence

The partial sums increase, because all the tex2html_wrap_inline589's were assumed to be positive.


If on the other hand a series satisfies--for some q>1-- the condition

displaymath583

then the series will diverge; it will go "off to infinity''.

Draw a picture similar to the one on the previous page to convince yourself that this is what will happen!


In fact, we can generalize the idea behind these two results slightly to obtain the classical ratio test for series. Here we do not insist anymore that all the tex2html_wrap_inline589 's are positive and we only compare the limit of the ratios of the absolute values of the tex2html_wrap_inline589 's to 1:

Ratio Test
  1. If tex2html_wrap_inline597 , then the series

    displaymath584

    is convergent.

  2. If tex2html_wrap_inline599 , then the series

    displaymath584

    is divergent.

  3. The test is inconclusive when the limit equals 1! (This will not bother us much when we consider power series!)


An Example.

It's time to exploit this for power series. Consider the series

displaymath601

We want to find out for what values of x the series converges. If we view this power series as a series of the form

displaymath602

then tex2html_wrap_inline615 , tex2html_wrap_inline617 , tex2html_wrap_inline619 and so forth. The general term will have the form

displaymath603

(Plug in tex2html_wrap_inline621 to see that this formula works!) Consequently the ratios are given by

displaymath604

Since

displaymath605

we obtain

displaymath606

What's next? Do you remember the question we are trying to answer? For what values of x does the power series converge! The ratio test tells us now that the series will converge as long as |x|<1. It also tells us that the series will diverge for |x|>1. That gives us a pretty complete picture about what's going on:

The biggest interval (it is always an interval!) where a power series is converging is called interval of convergence of the power series. The interval of convergence is always centered at the center of the power series. It is customary to call half the length of the interval of convergence the radius of convergence of the power series. In our example, the center of the power series is 0, the interval of convergence is the interval from -1 to 1 (note the vagueness about the end points of the interval), its length is 2, so the radius of convergence equals 1.


Another example.

Consider the power series

displaymath629


One more example.

Consider the power series

displaymath655


Try it yourself

Find the radii of convergence of the following power series:

Click on the problem to see the answer. Click here to continue.


Helmut Knaust

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