3. 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).

4. Prove thatconverges. 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). Compute the ratio and its limit with Note that

5. 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).

6. 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).

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