Convergence Tests Absolute Convergence Alternating Series Rearrangements Lecture 24 Section 11.4 Absolute and Conditional Convergence; Alternating Series Jiwen He Department of Mathematics, University of Houston [email protected] http://math.uh.edu/∼jiwenhe/Math1432 Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 24 April 10, 2008 1 / 16 Convergence Tests Absolute Convergence Alternating Series Rearrangements Basic Series that Converge or Diverge Basic Series that Converge X Geometric series: xk , if |x| < 1 X 1 p-series: , if p > 1 kp Basic Series that Diverge X Any series ak for which lim ak 6= 0 k→∞ X 1 p-series: , if p ≤ 1 kp Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 24 April 10, 2008 2 / 16 Convergence Tests Absolute Convergence Alternating Series Rearrangements Basic Series that Converge or Diverge Basic Series that Converge X Geometric series: xk , if |x| < 1 X 1 p-series: , if p > 1 kp Basic Series that Diverge X Any series ak for which lim ak 6= 0 k→∞ X 1 p-series: , if p ≤ 1 kp Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 24 April 10, 2008 2 / 16 Convergence Tests Absolute Convergence Alternating Series Rearrangements Basic Series that Converge or Diverge Basic Series that Converge X Geometric series: xk , if |x| < 1 X 1 p-series: , if p > 1 kp Basic Series that Diverge X Any series ak for which lim ak 6= 0 k→∞ X 1 p-series: , if p ≤ 1 kp Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 24 April 10, 2008 2 / 16 Convergence Tests Absolute Convergence Alternating Series Rearrangements Basic Series that Converge or Diverge Basic Series that Converge X Geometric series: xk , if |x| < 1 X 1 p-series: , if p > 1 kp Basic Series that Diverge X Any series ak for which lim ak 6= 0 k→∞ X 1 p-series: , if p ≤ 1 kp Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 24 April 10, 2008 2 / 16 Convergence Tests Absolute Convergence Alternating Series Rearrangements Basic Series that Converge or Diverge Basic Series that Converge X Geometric series: xk , if |x| < 1 X 1 p-series: , if p > 1 kp Basic Series that Diverge X Any series ak for which lim ak 6= 0 k→∞ X 1 p-series: , if p ≤ 1 kp Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 24 April 10, 2008 2 / 16 Convergence Tests Absolute Convergence Alternating Series Rearrangements Basic Series that Converge or Diverge Basic Series that Converge X Geometric series: xk , if |x| < 1 X 1 p-series: , if p > 1 kp Basic Series that Diverge X Any series ak for which lim ak 6= 0 k→∞ X 1 p-series: , if p ≤ 1 kp Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 24 April 10, 2008 2 / 16 Convergence Tests Absolute Convergence Alternating Series Rearrangements Basic Series that Converge or Diverge Basic Series that Converge X Geometric series: xk , if |x| < 1 X 1 p-series: , if p > 1 kp Basic Series that Diverge X Any series ak for which lim ak 6= 0 k→∞ X 1 p-series: , if p ≤ 1 kp Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 24 April 10, 2008 2 / 16 Convergence Tests Absolute Convergence Alternating Series Rearrangements Basic Series that Converge or Diverge Basic Series that Converge X Geometric series: xk , if |x| < 1 X 1 p-series: , if p > 1 kp Basic Series that Diverge X Any series ak for which lim ak 6= 0 k→∞ X 1 p-series: , if p ≤ 1 kp Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 24 April 10, 2008 2 / 16 Convergence Tests Absolute Convergence Alternating Series Rearrangements Basic Series that Converge or Diverge Basic Series that Converge X Geometric series: xk , if |x| < 1 X 1 p-series: , if p > 1 kp Basic Series that Diverge X Any series ak for which lim ak 6= 0 k→∞ X 1 p-series: , if p ≤ 1 kp Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 24 April 10, 2008 2 / 16 Convergence Tests Absolute Convergence Alternating Series Rearrangements Convergence Tests (1) Basic Test for Convergence Keep in Mind that, P if ak 9 0, then the series ak diverges; therefore there is no reason to apply any special convergence test. Examples P k P k k x with |x| ≥ 1 (e.g, (−1) ) diverge since x 9 0. X k diverges since k → 1 6= 0. k + 1 k+1 k X 1 k 1 − diverges since a = 1 − 1 → e−1 6= 0. k k k Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 24 April 10, 2008 3 / 16 Convergence Tests Absolute Convergence Alternating Series Rearrangements Convergence Tests (1) Basic Test for Convergence Keep in Mind that, P if ak 9 0, then the series ak diverges; therefore there is no reason to apply any special convergence test. Examples P k P k k x with |x| ≥ 1 (e.g, (−1) ) diverge since x 9 0. X k diverges since k → 1 6= 0. k + 1 k+1 k X 1 k 1 − diverges since a = 1 − 1 → e−1 6= 0. k k k Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 24 April 10, 2008 3 / 16 Convergence Tests Absolute Convergence Alternating Series Rearrangements Convergence Tests (1) Basic Test for Convergence Keep in Mind that, P if ak 9 0, then the series ak diverges; therefore there is no reason to apply any special convergence test. Examples P k P k k x with |x| ≥ 1 (e.g, (−1) ) diverge since x 9 0. X k diverges since k → 1 6= 0. k + 1 k+1 k X 1 k 1 − diverges since a = 1 − 1 → e−1 6= 0. k k k Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 24 April 10, 2008 3 / 16 Convergence Tests Absolute Convergence Alternating Series Rearrangements Convergence Tests (1) Basic Test for Convergence Keep in Mind that, P if ak 9 0, then the series ak diverges; therefore there is no reason to apply any special convergence test. Examples P k P k k x with |x| ≥ 1 (e.g, (−1) ) diverge since x 9 0. X k diverges since k → 1 6= 0. k + 1 k+1 k X 1 k 1 − diverges since a = 1 − 1 → e−1 6= 0. k k k Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 24 April 10, 2008 3 / 16 Convergence Tests Absolute Convergence Alternating Series Rearrangements Convergence Tests (1) Basic Test for Convergence Keep in Mind that, P if ak 9 0, then the series ak diverges; therefore there is no reason to apply any special convergence test. Examples P k P k k x with |x| ≥ 1 (e.g, (−1) ) diverge since x 9 0. X k diverges since k → 1 6= 0. k + 1 k+1 k X 1 k 1 − diverges since a = 1 − 1 → e−1 6= 0. k k k Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 24 April 10, 2008 3 / 16 Convergence Tests Absolute Convergence Alternating Series Rearrangements Convergence Tests (1) Basic Test for Convergence Keep in Mind that, P if ak 9 0, then the series ak diverges; therefore there is no reason to apply any special convergence test. Examples P k P k k x with |x| ≥ 1 (e.g, (−1) ) diverge since x 9 0. X k diverges since k → 1 6= 0. k + 1 k+1 k X 1 k 1 − diverges since a = 1 − 1 → e−1 6= 0. k k k Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 24 April 10, 2008 3 / 16 Convergence Tests Absolute Convergence Alternating Series Rearrangements Convergence Tests (1) Basic Test for Convergence Keep in Mind that, P if ak 9 0, then the series ak diverges; therefore there is no reason to apply any special convergence test. Examples P k P k k x with |x| ≥ 1 (e.g, (−1) ) diverge since x 9 0. X k diverges since k → 1 6= 0. k + 1 k+1 k X 1 k 1 − diverges since a = 1 − 1 → e−1 6= 0. k k k Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 24 April 10, 2008 3 / 16 Convergence Tests Absolute Convergence Alternating Series Rearrangements Convergence Tests (2) Comparison Tests Rational terms are most easily handled by basic comparison or limit comparison with p-series P 1/kp Basic Comparison Test X 1 X 1 converges by comparison with 2k3 + 1 k3 X k3 X 1 converges by comparison with k5 + 4k4 + 7 k2 X 1 X 2 converges by comparison with k3 − k2 k3 X 1 X 1 diverges by comparison with 3k + 1 3(k + 1) X 1 X 1 diverges by comparison with ln(k + 6) k + 6 Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 24 April 10, 2008 4 / 16 Convergence Tests Absolute Convergence Alternating Series Rearrangements Convergence Tests (2) Comparison Tests Rational terms are most easily handled by basic comparison or limit comparison with p-series P 1/kp Basic Comparison Test X 1 X 1 converges by comparison with 2k3 + 1 k3 X k3 X 1 converges by comparison with k5 + 4k4 + 7 k2 X 1 X 2 converges by comparison with k3 − k2 k3 X 1 X 1 diverges by comparison with 3k + 1 3(k + 1) X 1 X 1 diverges by comparison with ln(k + 6) k + 6 Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 24 April 10, 2008 4 / 16 Convergence Tests Absolute Convergence Alternating Series Rearrangements Convergence Tests (2) Comparison Tests Rational terms are most easily handled by basic comparison or limit comparison with p-series P 1/kp Basic Comparison Test X 1 X 1 converges by comparison with 2k3 + 1 k3 X k3 X 1 converges by comparison with k5 + 4k4 + 7 k2 X 1 X 2 converges by comparison with k3 − k2 k3 X 1 X 1 diverges by comparison with 3k + 1 3(k + 1) X 1 X 1 diverges by comparison with ln(k + 6) k + 6 Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 24 April 10, 2008 4 / 16 Convergence Tests Absolute Convergence Alternating Series Rearrangements Convergence Tests (2) Comparison Tests Rational terms are most easily handled by basic comparison or limit comparison with p-series P 1/kp Basic Comparison Test X 1 X 1 converges by comparison with 2k3 + 1 k3 X k3 X 1 converges by comparison with k5 + 4k4 + 7 k2 X 1 X 2 converges by comparison with k3 − k2 k3 X 1 X 1 diverges by comparison with 3k + 1