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Unformatted text preview: is a positive constant, the
power series
x
x
1+ +
b
b 2 3 x
+
b ∞ + ··· =
n=0 x
b n (1.2) has radius of convergence b. To see this, we make the substitution y = x/b,
∞
and the power series becomes n=0 y n , which we already know converges for
y  < 1 and diverges for y  > 1. But
y  < 1 ⇔ y  > 1 ⇔ x
<1
b
x
>1
b ⇔ x < b, ⇔ x > b. 1 Good references for the theory behind convergence of power series are Edward D.
Gaughan, Introduction to analysis , Brooks/Cole Publishing Company, Paciﬁc Grove, 1998
and Walter Rudin, Principles of mathematical analysis , third edition, McGrawHill, New
York, 1976. 3 Thus for x < b the power series (1.2) converges to
1
1
b
=
=
,
1−y
1 − (x/b)
b−x
while for x > b, it diverges.
There is a simple criterion that often enables one to determine the radius of
convergence of a power series.
Ratio Test. The radius of convergence of the power series
∞ a0 + a1 (x − x0 ) + a2 (x − x0 )2 + · · · = an (x − x0 )n
n=0 is given by the formula
R = lim n→∞ an 
,
an+1  so long as t...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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