This post will not answer the question – which would be inappropriate because I am a judge – but rather, will address the issue of the convergence of the series: (*) for . is necessary condition for the series: x₁ + x₂ + x₃ + ...converging. Therefore, series * does not converge when x = -1 or x = 1. Clearly, series * does converge when x = 0. Finally, please consider 0 < |x| < 1. We may apply the http://www.shu.edu/projects/reals/numser/t_alter.html

 for –1 < x < 0, and the http://mathworld.wolfram.com/RootTest.html

,for 0 < x < 1, to determine series * does converge, for|x| < 1. Alternating series test: Root test: (It is easy to show, via http://www-math.mit.edu/~djk/18_01/chapter26/contents.html

 .)

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