Power Series

In Glogpedia

by stephaniejoy19
Last updated 6 years ago


Toggle fullscreen Print glog
Power Series

Q: What if the expression obtained from the ratio test is 0*|x-a|?A: Then the expression is always less than 1, so the series converges anywhere, for any value of x.Q: What if the expression obtained from the ratio test is infinity*|x-a|?A: Then the series diverges everywhere except at x=a.

The video above shows the procedure in action!

Power Series

So...the series isn't a series of numbers, but of expressions that depend on the variable x.

Here is the general form for a power series.

But what do we do with series that look like this?

To Find an Interval of Convergence:* Use the ratio test to begin.* Set the expression obtained from the ratio test to be less than 1 and solve for x. For these values of x, we know the series will converge.*Now, don't forget to test the endpoints. Plug each endpoint in for x in the original series and test for convergence/divergence. If the series converges for an endpoint, then that endpoint is included in the interval of convergence.

The first thing we usually want to do with a power series is to find the interval of convergence.

That is, find the values of x for which the series will converge.

On the interval of convergence, we input a value for x and get a new value - the value to which the series converges. So really, it is a function!

Soon, you will take this knowledge of Power Series and extend it to be able to represent functions we already know as an infinite series called the Taylor Series.


    There are no comments for this Glog.