## Transcribed Text

1
show that the
series W n=1 (-1)" converges conditionally (Its sum is - -In2).
00
What about Et"?
n=1
2
suppose that a series W as *[n]7"
n=-00
converges to an analytic function X(Z) in some annulus R1 < 1zl < Rz - That sum X(Z)
is called the Z-transform of X[n] (n=0,+1.12,
)
use this
expression f(Z)
(n = 0,I1, +2, )
for the coefficients in Laurent series to show that if the annulus contains
the unit circle 1Z1= 1, , then the inverse Z-transform of X(Z) can be written
TT
= 2TT e do
n=0,+1,52
3
Compute of
the Laurent expansion F(Z)= (Z-1)(2-3) 1 + Z-5 Z2 avound Zo=0 in
each of the regions 1ZK<, 1 <12/<3, 3<121<5 and 1z175
4
Determine the radius of convergence of the Taylor series around Zo=0 of
a
F(Z) = sin (COSZ)
b
F(Z) = sin(tan Z)
C
F(Z)= Z+ 2 Z + 1
Z'-Z-1
5
Find the Taylor series Coefficients UP to and including the 2 term around ZEO
of
a
F(Z) = sin(tanz)
b
F(Z) = Z2+1
2²-2-1
C
F(Z) = tan(sinz)

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction
of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice.
Unethical use is strictly forbidden.