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1 arn Since the series is alternating, we use the Alternating Series Test. Easy comparison can be made to the p-series. 3. Limit Comparison Test. Advantages: This test is easier to apply than the comparison test. Disadvantages: Same An alternative proof of the convergence of the p-series for p > 1 and divergence for 0 < p ≤ 1, using the integral test, is given in Example 12.44. Page 10.
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A p -series converges when p > 1 and diverges when p < 1. Here are a few important examples of p -series that are either convergent or divergent. A commonly-used corollary of the integral test is the p-series test.
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For instance, the sum of the example series is . If p=1, we call the resulting series the harmonic series: By the above theorem, the harmonic series does not converge. From these estimates, we have the following test for the p-series: Theorem. The p-series is divergent when p ≤ 1, and in this case, lim n→∞ s 2n(p) s n(p) = 2 2p. (1) The p-series is convergent for p > 1, and in this case, 2p −1 2 p−2 ≤ lim n→∞ s n(p) ≤ 2p 2 −2.
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is convergent if p > 1 and is divergent if p 1. Discussion [Using Flash] Estimating using the Integral Test. The \(N\)th term test, generally speaking, does not guarantee convergence of a series.Convergence or divergence of a series is proved using sufficient conditions.The comparison tests we consider below are just the sufficient conditions of convergence or divergence of series.
One often compares to a p-series when using the Comparison Test. Example. Test the series P 1 n=1 1 2+3 for convergence. Solution.
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If p is less than 1, the series diverges. P series or P-series may refer to: the p-series in mathematics; P-series fuels, blends of fuels; Huawei P series, mobile phone series by Huawei; IBM pSeries, computer series by IBM; Ruger P series – pistols; Sony Cybershot P-series digital cameras, see Cyber-shot; Sony Vaio P series – notebook computers; Sony Ericsson P series, a series of cell phones; Vespa P-series motor scooters; See also The p -series is a power series of the form or, where p is a positive real number and k is a positive integer. The p -series test determines the nature of convergence of a p -series as follows: The p -series converges if and diverges if. See more Calculus topics Videos related to Calculus The p-series is given by 1/n p = 1/1 p + 1/2 p + 1/3 p + where p > 0 by definition.