For a series , let . If , choose ; eventually , so geometric comparison gives absolute convergence. If , the terms fail to tend to zero along a subsequence and the series diverges. At , the test is inconclusive.
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The Root Test is a method used to determine the convergence or divergence of an infinite series. Specifically, it helps assess the behavior of a series of the form: \[ \sum_{n=1}^{\infty} a_n \] where \( a_n \) is a sequence of real or complex numbers. The primary approach is based on the concept of the \( n \)-th root of the absolute value of the terms in the series.