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Some an sequences are arithmetic some are geometric

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Unformatted text preview: artial fraction decomposition. Theorem 8.11. Suppose an xn + an−1 xn−1 + · · · + a2 x2 + a1 x + a0 = bm xm + mm−1 xm−1 + · · · + b2 x2 + b1 x + b0 for all x in an open interval I . Then n = m and ai = bi for all i = 1 . . . n. Believe it or not, the proof of Theorem 8.11 is a consequence of Theorem 3.14. Define p(x) to be the difference of the left hand side of the equation in Theorem 8.11 and the right hand side. Then p(x) = 0 for all x in the open interval I . If p(x) were a nonzero polynomial of degree k , then, by Theorem 3.14, p could have at most k zeros in I , and k is a finite number. Since p(x) = 0 for all the x in I , p has infinitely many zeros, and hence, p is the zero polynomial. This means there can be no nonzero terms in p(x) and the theorem follows. Arguably, the best way to make sense of either of the two preceding theorems is to work some examples. Example 8.6.1. Resolve the following rational functions into partial fractions. 1. R(x) = 2. R(x) = x+5 −x−1 3. R(...
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