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Series such as

Series such as. arise in applications, but the convergence tests developed so far cannot be applied easily. Fortunately, the Ratio Test can be used for this and many other series. THEOREM 1 Ratio Test Assume that the following limit exists:. ( i ) If ρ < 1, then.

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Series such as

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  1. Series such as arise in applications, but the convergence tests developed so far cannot be applied easily. Fortunately, the Ratio Test can be used for this and many other series. THEOREM 1 Ratio Test Assume that the following limit exists: (i) If ρ < 1, then converges absolutely. (ii) If ρ > 1, then diverges. (iii) If ρ = 1, the test is inconclusive (the series may converge or diverge).

  2. converges. Prove that THEOREM 1 Ratio Test Assume that the following limit exists: (i) If ρ < 1, then converges absolutely. (ii) If ρ > 1, then diverges. (iii) If ρ = 1, the test is inconclusive (the series may converge or diverge). Note that Compute the ratio and its limit with

  3. Does converge? THEOREM 1 Ratio Test Assume that the following limit exists: (i) If ρ < 1, then converges absolutely. (ii) If ρ > 1, then diverges. (iii) If ρ = 1, the test is inconclusive (the series may converge or diverge).

  4. Does converge? THEOREM 1 Ratio Test Assume that the following limit exists: (i) If ρ < 1, then converges absolutely. (ii) If ρ > 1, then diverges. (iii) If ρ = 1, the test is inconclusive (the series may converge or diverge).

  5. Ratio Test Inconclusive Show that both convergence and divergence are possible when ρ = 1 by considering For an = n2, we have On the other hand, for bn = n-2, diverges and Thus, ρ = 1 in both cases, but converges. This shows that both convergence and divergence are possible when ρ = 1.

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