Pde

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Unformatted text preview: , n! n=0 ... As the number of terms increases, the sum approaches the familiar value of the exponential function ex at x = 1. For a power series to be useful, the infinite sum must actually add up to a finite number, as in this example, for at least some values of the variable x. We let sN denote the sum of the first (N + 1) terms in the power series, N sN = a0 + a1 (x − x0 ) + a2 (x − x0 )2 + · · · + aN (x − x0 )N = an (x − x0 )n , n=0 and say that the power series ∞ an (x − x0 )n n=0 converges if the finite sum sN gets closer and closer to some (finite) number as N → ∞. Let us consider, for example, one of the most important power series of applied mathematics, the geometric series ∞ 1 + x + x2 + x3 + · · · = xn . n=0 In this case we have sN = 1 + x + x2 + x3 + · · · + xN , xsN = x + x2 + x3 + x4 · · · + xN +1 , sN − xsN = 1 − xN +1 , sN = 1 − xN +1 . 1−x If |x| < 1, then xN +1 gets smaller and smaller as N approaches infinity, and hence lim xN +1 = 0. N →∞ Substituting into the expression for sN , we find that lim sN = N →∞ 1 . 1−x Thus if |x| < 1, we say that the geometric se...
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This document was uploaded on 01/12/2014.

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