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Unformatted text preview: er series for ex , and hence must also be
inﬁnite.
Power series with positive radius of convergence are so important that there
is a special term for describing functions which can be represented by such power
series. A function f (x) is said to be real analytic at x0 if there is a power series
∞ an (x − x0 )n
n=0 about x0 with positive radius of convergence R such that
∞ an (x − x0 )n , f (x) = for n=0 5 x − x0  < R. For example, the functions ex is real analytic at any x0 . To see this, we
utilize the law of exponents to write ex = ex0 ex−x0 and apply (1.1) with x
replaced by x − x0 :
∞ ex = ex0 ∞ 1
an (x − x0 )n ,
(x − x0 )n =
n!
n=0
n=0 where an = ex0
.
n! This is a power series expansion of ex about x0 with inﬁnite radius of convergence. Similarly, the monomial function f (x) = xn is real analytic at x0
because
n
n!
xn = (x − x0 + x0 )n =
xn−i (x − x0 )i
i!(n − i)! 0
i=0
by the binomial theorem, a power series about x0 in which all but ﬁnitely many
of the coeﬃcients are zero...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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