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Determining Analytically the DTFT of Sequences

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Determine analytically the DTFT of each of the following sequences. See attachment for details

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Step by step calculations of DTFT of sequences are included. Geometric series theory, time shift theorem, and Euler's theorem are applied in the solution.

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Determine analytically the DTFT of each of the following sequences.
DTFT of y(k) = Y{e^(jw)} = Sigma(from k = -~ to k = +~) [ y(k)*e^-(jwk) ]
(where Sigma is the summation, "~" is infinity, and "^" is the exponent (power) operator)
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1. y(k) = 3(0.9)^k u(k)
Y{e^(jw)} = Sigma(from k = -~ to k = +~) [ 3(0.9)^k u(k)*e^-(jwk) ]
= 3* Sigma(from k = 0 to k = +~) ...

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