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123 PL K LLL What functions are represented by the following power series a kzk...

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123 PL K LLL What functions are represented by the following power series a kzk b k k k 1 Solution a k 0 Multiply by z obtain Differentiate the geometric series 1 k 0 k 1 1 We get zk Multiply by 22 obtain k 1 k 1 k k 1 k 2 b Differentiate the geometric series oz twice obtain k 0 02 obtain k 0 kzk 1 kzk k 1 2 1 2 1 1 z k k 1 1 z k k 1 zk 2 k 2 22 1 z 22 2 1 k 2 2 1 2 k k 1 z k 2 k 2 z 22 1 z 2 1 z

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