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Lesson 26a - Power Series

Lesson 26a - Power Series - PowerSeries .Letuslookatan...

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Power Series Imagine you had a functions that was very difficult to integrate. Let us look at an example: 2 ) ( x e x f One could prove that this has no nice closed form integral with the standard functions we have come to use and love. So then how could we integrate this? Perhaps it does not matter, this function may not even really appear in real life right? Let us take a look at this function: Wait… this looks like a normal curve… which is used a lot in different areas….
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Power Series So we know we would like to work with these tough functions, but integrating them is tough, so what could we do? Wouldn’t it be nice if there was a way to change a function into a polynomial? (Hint there is!!!) This idea is the premise that led to power series: power series is a series that is a function of x that looks like: A power series is a series that is a function of x that looks like: ... ) ( ) ( ) ( ) ( 3 3 2 2 1 0 0 a x a a x a a x a a a x a n n n Here a n is some expression that does not contain our variable x but will (almost always) depend on n, and a is some constant (known as the centre). We will be looking at how we can turn regular functions into power series, and how we can use them to solve tougher questions!
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Radius/Interval of Convergence When we have a power series, we need to know which x values we can substitute into the infinite series and still have a convergent series. Interval of Convergence: The interval of convergence is the set of x values that can be substituted into the power series, and the series will converge. Radius of Convergence: e radius of convergence is half of the length of the interval of convergence (similar The radius of convergence is half of the length of the interval of convergence (similar to that of a circle). entre: Centre: The centre of the radius will always be “a” in our definition of a power series: ... ) ( ) ( ) ( ) ( 3 2 a x a a x a a x a a a x a n
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Lesson 26a - Power Series - PowerSeries .Letuslookatan...

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