Power Series

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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.


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