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MATH231 Lecture Notes 7 - Power Series 8.6 Power series are...

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Unformatted text preview: Power Series: 8.6 Power series are one of the main reasons for studying series. Many of the elementary functions from precalc and calculus (sines and cosines, exponentials, logarithms, etc.) can be represented as a power series. A power series is (roughly speaking) a polynomial of infinite degree. More precisely a power series is a series of the following form ∞ summationdisplay k =0 b k ( x- c ) k . The constants b k are referred to as the coefficients of the power series, and we frequently say that the series above is centered at c . Example 1: The following are all power series ∑ ∞ k =0 x k k ! ∑ ∞ k =0 ( − 1) k ( x − 1) 2 k k ! ∑ ∞ k =0 (1 + x ) k 2 ∑ ∞ k =0 k ! x k As the are all sums of (integer) powers of x- c multiplied by some coefficient. So things that are NOT power series are ∑ ∞ k =0 ( x − k ) k k ! ∑ ∞ k =0 ( − 1) k ( x ) k sin( kx ) k ! ∑ ∞ k =0 sin( kx ) k The last is something called a Fourier series which we may discuss a little in...
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MATH231 Lecture Notes 7 - Power Series 8.6 Power series are...

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