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Section 5.3
Question 2: The limit of the ratios of the terms in \sum a_j (zz_0)^j is L zz_0, because of the given
information that lim_j a_{j+1} / a_j = L. Thus, by the ratio test, the series converges when zz_0 < 1/L and
diverges when zz_0 > 1/L. Thus the radius of convergence of the series is 1/L.
Question 3 (b). L = 2, so the radius of convergence is 1/2, and the circle of convergence is z  1 = 1/2.
(d) L = 1/3, so the radius of convergence is 3, and the circle of convergence is z  i = 3.
Section 5.5
Question 3: Although it is not mentioned explicitly, it is assumed that we are computing the Laurent series
aroud zero.
The easiest way is to break the function up into partial fractions:
z/(z+1)(z2) = (1/3)/(z+1) + (2/3)/(z2);
You can either work out the constants 1/3, 2/3 by the usual method taught to you in lowerdivision, or you
can compute the residues of z/(z+1)(z2) at 1 and 2 respectively.
(a) When z<1 we have
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 Spring '08
 Grossman
 Ratios

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