9.5_LA-Solutions-Ratio_Test_and_Convergence_Tests

9.5_LA-Solutions-Ratio_Test_and_Convergence_Tests - D^ Z d...

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MthSc 108-9.5 (Ratio Test & Convergence Tests) Solutions 1. Apply the Ratio Test to the series 1 n 3 n = 1 . What is your conclusion about the convergence/divergence about this series based on your results of the Ratio Test? What other test might you use? Observe that the terms of the series are positive and that the ratio test can be applied. Compare this series to the convergent p-series (p=3>1), 1 n 3 n = 1 . lim n →∞ a n + 1 a n = lim n →∞ 1 ( n + 1) 3 1 n 3 = lim n →∞ n 3 ( n + 1) 3 = 1 Since ρ = 1, the Ratio Test applied to this series is inconclusive. However, it is easy to see that 1 n 3 n = 1 is a convergent p-series where p = 3 > 1 . 2. Use the Limit Comparison Test to determine if the series 1 2 n 2 + n n = 1 converges or diverges. Compare this series to the convergent p-series, 1 n 2 n = 1 , where p = 2 > 1 . lim n →∞ a n b n = lim n →∞ 1 n 2 1 2 n 2 + n = lim n →∞ 2 n 2 + n 1 1 n 2 == lim n →∞ 2 + 1 n
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9.5_LA-Solutions-Ratio_Test_and_Convergence_Tests - D^ Z d...

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