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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
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.
- Winter '14