On, off, on, off

Logic Level 2

A lamp is turned off and on by a push-button switch. Starting at time zero, the lamp is on for \(\frac{1}{2}\) minute, then turned off for \(\frac{1}{4}\) minute, then on for \(\frac{1}{8}\) minute, and so on. Knowing that \(\frac{1}{2}\) + \(\frac{1}{4}\) + \(\frac{1}{8}\) \(+ ...\) \(= 1\), and that the button is continuously working, will the lamp be turned on or off after one minute? Assume that the lamp would neither be burnt out nor explode in the process.

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