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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)

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