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Unformatted text preview: his limit exists.
Let us check that the ratio test gives the right answer for the radius of convergence of the power series (1.2). In this case, we have
an = 1
,
bn so an 
bn+1
1/bn
= n = b,
=
an+1 
1/bn+1
b and the formula from the ratio test tells us that the radius of convergence is
R = b, in agreement with our earlier determination.
In the case of the power series for ex ,
∞ 1n
x,
n!
n=0
in which an = 1/n!, we have
an 
1/n!
(n + 1)!
=
=
= n + 1,
an+1 
1/(n + 1)!
n!
and hence
R = lim n→∞ an 
= lim (n + 1) = ∞,
an+1  n→∞ so the radius of convergence is inﬁnity. In this case the power series converges
for all x. In fact, we could use the power series expansion for ex to calculate ex
for any choice of x.
On the other hand, in the case of the power series
∞ n!xn ,
n=0 4 in which an = n!, we have
an 
n!
1
=
=
,
an+1 
(n + 1)!
n+1 an 
= lim
an+1  n→∞ R = lim n→∞ 1
n+1 = 0. In this case, the radius of convergence is zero, and the power series does not
conver...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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