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Understanding the Ratio Test for Series Convergence and Divergence

The Ratio Test is a powerful tool in determining the convergence of infinite series. This theorem states that for a given limit, if ρ < 1, the series converges absolutely; if ρ > 1, the series diverges; and if ρ = 1, the test is inconclusive. This guide explores the Ratio Test's applications, limitations, and includes examples to illustrate scenarios where series may converge or diverge when ρ is equal to 1. Understanding these principles is crucial for mastering series analysis in calculus.

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Understanding the Ratio Test for Series Convergence and Divergence

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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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