Section 5.1
Question 1(b): [I use \sum to denote the Sigma summation sign] This is 3 \sum z^k with z = 1/(1+i), so by the
geometric series formula and the fact that z < 1, we obtain 3/(1  z) = 3/(1  1/(1+i)) = 3  3i.
(c) Similarly, the expression is \sum z^k with z = 2/3, so we obtain 1 / (1  z) = 3/5.
Question 3: If z_n converges to z (say), then z_{n1} also converges to z. So lim (z_n  z_{n1}) = lim z_n 
lim z_{n1} = z  z = 0.
Question 7: (ae) use the Ratio test and find the limit of z_{n+1}/z_n, using the same techniques you would
use in realvariable calculus. For (f), note that the summands do not go to zero (in fact  i^k  1/k^2 
converges to 1, not zero), so the series has to diverge.
Section 5.2
Question 1(a): If f(z) = exp(z), then the j^th derivative of f(z) is exp(z) if j is even and exp(z) if j is odd.
Thus f^(j)(0) = (1)^j, and the result follows by Taylor's formula (at z_0 = 0).
(e): If f(z) = sinh(z), then the j^th derivative of f(z) is sinh(z) if j is even and cosh(z) if j is odd. Thus f^(2j)(0)
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 Spring '08
 Grossman
 Geometric Series

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