MAT137 Course website http://uoft.me/MAT137 . Basic comparison test and limit comparison test Watch Videos 13.2–13.4 before next Monday’s class. Supplementary video: 13.1 Watch Videos 13.5–13.9 before next Wednesday’s class. Francisco Guevara Parra () MAT137 27 February 2019 1 / 11 A simple BCT application ∞ 1 dx We want to determine whether x Z1 x + e is convergent or divergent. We can try at least two comparisons: 1 1 1. Compare and . x x + ex 1 1 2. Compare and . ex x + ex Try both. What can you conclude from each one of them? Francisco Guevara Parra () MAT137 27 February 2019 2 / 11 True or False Let a R. ∈ Let f and g be continuous functions on [a, ). ∞ Assume that x a, 0 f (x) g(x) . ∀ ≥ ≤ ≤ What can we conclude? ∞ ∞ 1. IF f (x)dx is convergent, THEN g(x)dx is convergent. Za Za ∞ ∞ 2. IF f (x)dx = , THEN g(x)dx = . Za ∞ Za ∞ ∞ ∞ 3. IF g(x)dx is convergent, THEN f (x)dx is convergent. Za Za ∞ ∞ 4. IF g(x)dx = , THEN f (x)dx = . Za ∞ Za ∞ Francisco Guevara Parra () MAT137 27 February 2019 3 / 11 True or False - Part II Let a R. ∈ Let f and g be continuous functions on [a, ). ∞ Assume that x a, f (x) g(x) . ∀ ≥ ≤ What can we conclude? ∞ ∞ 1. IF f (x)dx is convergent, THEN g(x)dx is convergent. Za Za ∞ ∞ 2. IF f (x)dx = , THEN g(x)dx = . Za ∞ Za ∞ ∞ ∞ 3. IF g(x)dx is convergent, THEN f (x)dx is convergent. Za Za ∞ ∞ 4. IF g(x)dx = , THEN f (x)dx = . Za ∞ Za ∞ Francisco Guevara Parra () MAT137 27 February 2019 4 / 11 What can you conclude? Let a R. Let f be a continuous, positive function on [a, ). ∈ ∞ ∞ In each of the following cases, what can you conclude about f (x)dx? Za Is it convergent, divergent, or we do not know? b 1. b a, M R s.t. f (x)dx M. ∀ ≥ ∃ ∈ Za ≤ b 2. M R s.t. b a, f (x)dx M. ∃ ∈ ∀ ≥ Za ≤ 3. M > 0 s.t. x a, f (x) M. ∃ ∀ ≥ ≤ 4. M > 0 s.t. x a, f (x) M. ∃ ∀ ≥ ≥ Francisco Guevara Parra () MAT137 27 February 2019 5 / 11 BCT calculations Use the BCT to determine whether each of the following is convergent or divergent 2 2 ∞ 1+cos x ∞ arctan x dx dx 1. 2/3 3. x Z1 x Z0 1+ e 2 ∞ 1+cos x ∞ x 2 dx 4. e− dx 2. 4/3 Z1 x Z0 (ln x)10 ∞ dx 5. 2 Z2 x Francisco Guevara Parra () MAT137 27 February 2019 6 / 11 Quick review For which values of p R is each of the following ∈ improper integrals convergent? 1 ∞ dx 1. p Z1 x 1 1 dx 2. p Z0 x 1 ∞ dx 3. p Z0 x Francisco Guevara Parra () MAT137 27 February 2019 7 / 11 A variation on LCT This is the theorem you have learned: Theorem (Limit-Comparison Test) Let a R. Let f and g be positive, continuous functions on [a, ). ∈ ∞ f (x) IF the limit L = lim exists and L > 0. x→∞ g(x) ∞ ∞ THEN f (x)dx and g(x)dx Za Za are both convergent or both divergent. Francisco Guevara Parra () MAT137 27 February 2019 8 / 11 A variation on LCT This is the theorem you have learned: Theorem (Limit-Comparison Test) Let a R. Let f and g be positive, continuous functions on [a, ). ∈ ∞ f (x) IF the limit L = lim exists and L > 0. x→∞ g(x) ∞ ∞ THEN f (x)dx and g(x)dx Za Za are both convergent or both divergent. What if we change the hypotheses to L = 0? Francisco Guevara Parra () MAT137 27 February 2019 8 / 11 A variation on LCT This is the theorem you have learned: Theorem (Limit-Comparison Test) Let a R. Let f and g be positive, continuous functions on [a, ). ∈ ∞ f (x) IF the limit L = lim exists and L > 0. x→∞ g(x) ∞ ∞ THEN f (x)dx and g(x)dx Za Za are both convergent or both divergent. What if we change the hypotheses to L = 0? 1. Write down the new version of this theorem (different conclusion). Francisco Guevara Parra () MAT137 27 February 2019 8 / 11 A variation on LCT This is the theorem you have learned: Theorem (Limit-Comparison Test) Let a R. Let f and g be positive, continuous functions on [a, ). ∈ ∞ f (x) IF the limit L = lim exists and L > 0. x→∞ g(x) ∞ ∞ THEN f (x)dx and g(x)dx Za Za are both convergent or both divergent. What if we change the hypotheses to L = 0? 1. Write down the new version of this theorem (different conclusion). 2. Prove it. Francisco Guevara Parra () MAT137 27 February 2019 8 / 11 A variation on LCT This is the theorem you have learned: Theorem (Limit-Comparison Test) Let a R. Let f and g be positive, continuous functions on [a, ). ∈ ∞ f (x) IF the limit L = lim exists and L > 0. x→∞ g(x) ∞ ∞ THEN f (x)dx and g(x)dx Za Za are both convergent or both divergent. What if we change the hypotheses to L = 0? 1. Write down the new version of this theorem (different conclusion). 2. Prove it. f (x) Hint: If lim = 0, what is larger f (x) or g(x)? x→∞ g(x) Francisco Guevara Parra () MAT137 27 February 2019 8 / 11 Convergent or divergent? 3 ∞ x +2x +7 1. dx Z x 5 + 11x 4 +1 1 sin x 1 dx 5. 3/2 ∞ 1 Z x 2. dx 0 2 Z1 √x + x +1 ∞ x 2 1 6. e− dx 3cos x Z 3. dx 0 Z x x 0 + √ 10 ∞ (ln x) 1 dx 7. 2 4. cot x dx Z2 x Z0 Francisco Guevara Parra () MAT137 27 February 2019 9 / 11 Comparison tests for type 2 improper integrals ∞ A type-1 improper integral is an integral of the form f (x)dx, Za where f is a continuous, bounded function on [c, ) ∞ b A type-2 improper integral is an integral of the form f (x)dx, Za where f is a continuous function on (a, b] possibly with a vertical asymptote at a. In the videos, you learned BCT and LCT for type-1 improper integrals. 1. Write the statement of the BCT for type-2 improper integrals. 2. Write the statement of the LCT for type-2 improper integrals. 3. Construct an example that you can prove is convergent thanks to one of these theorems. 4. Construct an example that you can prove is divergent thanks to one of these theorems. Francisco Guevara Parra () MAT137 27 February 2019 10 / 11.