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notes13 - Functions as Power Series Last time we began the...

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Functions as Power Series. Last time we began the study of Power Series and the values for which they converge. In summary, we looked at the function f ( x ) = X k =0 a k ( x - a ) k and asked, “what is the domain of this function?”. The answer was, every x that is inside the interval of convergence for this power series. Math 9C Summer 2011 (UCR) Pro-Notes October 24, 2011 1 / 1
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For some power series, we saw there was only one x which the power series converged (This was x = a ). For other we saw that it could be an interval of the form ( a - R , a + R ) or even the entire real line ( -∞ , ). So now we ask: if we define the function f ( x ) = X k =0 a k ( x - a ) k , have we ever seen this function before? For some power series the answer is a partial yes. Our first example of a power series was the geometric series X k =0 x k we know this power series has an interval of convergence ( - 1 , 1). Math 9C Summer 2011 (UCR) Pro-Notes October 24, 2011 2 / 1
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If we define the function f ( x ) = X k =0 x k , then the domain of this function is ( - 1 , 1). But we can do better, we know that when - 1 < x < 1, X k =0 x k = 1 1 - x . Something to think about: We can put 2 into the right side of this equation, but we can’t put 2 into the left side of this equation. So this equation is only true for - 1 < x < 1. So we say in words that
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