Write a power series for the function centered at c
It's going to be equal to-- if it converges. This will actually, I guess, give us a sensical answer. And we can expand it out.
This one, of course, will simplify just a sub 0. This idea of convergence is important here. Example 3 Find a power series representation for the following function and determine its interval of convergence. We're multiplying by the same thing over here.
Our common ratio in this example is x. And this whole idea of using power series, or in this special case, geometric series to represent functions, has all sorts of applications in engineering and finance. We're defining a function in terms of x.
And of course, we don't have to use x all the time as the independent variable, but this is kind of the most typical convention.
So we could say that our radius of convergence is equal to 1. So let me write that down.
Going from one term to the next, we're just multiplying by x. And the most common one that you will see in your mathematical careers is the power series.
Power series representation of polynomials
And the most common one that you will see in your mathematical careers is the power series. Now, what's exciting about this is we know that this, under certain conditions, will actually give us a finite value. As long as x stays within one of 0, and that's the same thing as saying this right over here, this series is going to converge. I guess we could also use r as an independent variable if we wanted as well. And we know what that finite sum is. So as long as our x value stays less than a certain amount from our c value, then this thing will converge. You could almost imagine this is shifting our function to the n-th power. Our common ratio in this example is x. So, hopefully we now have an idea on how to find the power series representation for some functions. So let's just think about defining a function in terms of a geometric series. So let me write that down. It can't go all the way to 1, but as long as it stays less than 1, or as long as it stays greater than negative 1.
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