AP Calc BC Convergence Tests Name: Block: Seat: n th Term Divergence Test n=k diverges if lim n a n 0 a n diverges if lim n a n does not exist 1. Determine the convergence n 1 n + 1 Geometric Series The series ar n n=0 converges only if 1 < r < 1 the sum is a 1 r 1. Determine the convergence ( 3 5 ) n if it converges, find the sum St. Francis High School AP Calc BC
Page 2 12 2015 p Series Test The series diverges if p 1 converges if p > 1 n=k 1 n p 1. Determine the convergence 3 n n Direct Comparison Test This test is used when a known series is bigger than the given series. If a n has no negative terms, and a ceiling function k b n converges, then k a n must also converge. If a n has no negative terms, and a floor function k b n diverges, then k a n must also diverge. 1. Use the direct Comparison Test to determine the convergence 1 n 2 + 3 St. Francis High School Page 2 12 AP Calc BC
Page 3 12 2015 2. Use the direct Comparison Test to determine the convergence 1 3 n + 2 3. Use the direct Comparison Test to determine the convergence 1 n 1/2 + 2 St. Francis High School Page 3 12 AP Calc BC
Page 4 12 2015 Limit Comparison Test This is one the most useful tests for determining convergence. Suppose a n > 0 and b n > 0 for all n > N where N is a positive integer. If lim n a n b n = c, 0 < c <, then a n and b n behave the same. 2. Use the Limit Comparison Test to determine the convergence 1 2 n 1 If lim n a n b n = 0, and b n converges, then a n converges. If lim n a n b n =, and b n diverges, then a n diverges. 1. Use the Limit Comparison Test to determine the convergence 3n + 2 (n + 1) 2 St. Francis High School Page 4 12 AP Calc BC
Page 5 12 2015 3. Use the Limit Comparison Test to determine the convergence 2n + 1 n 3 2n 4. Use the Limit Comparison Test to determine the convergence ln n n 3/2 St. Francis High School Page 5 12 AP Calc BC
Page 6 12 2015 Alternating Series Test for Convergence The series ( 1) n+1 a n = a 1 a 2 + a 3 a 4 +... 2. Use the AST to determine to determine the convergence 3( 1) n n 2 + 1 will converge if all three true: Each Term is positive The terms are eventually decreasing lim n a n = 0 1. Use the AST to determine to determine the convergence ( 1) n n St. Francis High School Page 6 12 AP Calc BC
Page 7 12 2015 3. Use the AST to determine to determine the convergence 10( 1) n n! 4. Use the AST to determine to determine the convergence ( 1) n 2 n e n 1 St. Francis High School Page 7 12 AP Calc BC
Page 8 12 2015 Root Test The next test for convergence or divergence series works especially well for series involving nth powers. Let a n be a series. n=k n If lim an = L exists, then n If L < 1, then the series converges 2. Use the Root test to determine the convergence 3 n e n If L > 1 then the series diverges If L = 1 then the test is inconclusive 1. Use the Root test to determine the convergence [ n 2 ] n + 1 2n 2 + 1 St. Francis High School Page 8 12 AP Calc BC
Page 9 12 2015 3. Use the Root test to determine the convergence e 3n n n 4. Use the Root test to determine the convergence [ 1 ] n arctan n St. Francis High School Page 9 12 AP Calc BC
Page 10 12 2015 Ratio Test This test is used most ten on the AP Exam to determine convergence a power series. Let n=k a n be a series with positive terms and lim a n+1 n a n = L 2. Use the Ratio Test to determine the interval and radius convergence (the values x that will make this converge): n=0 nx n 10 n If L < 1, then the series converges If L > 1 then the series diverges If L = 1 then the test is inconclusive 1. Use the Ratio Test to determine if the following converges: 2 n 3 n + 1 n=0 St. Francis High School Page 10 12 AP Calc BC
Page 11 12 2015 3. Use the Ratio Test to determine the interval and radius convergence (the values x that will make this converge): (x + 5) n n=0 2. Find the interval convergence for ( ) x ( 1) n+1 2n 2n After the Ratio Test find the values x that will make the limit less than 1. This interval values x that make it converge is the interval convergence. Be sure to check the end points the interval 1. Find the interval convergence for n=4 nx n 4 n (n 2 + 1) St. Francis High School Page 11 12 AP Calc BC
Page 12 12 2015 3. Find the interval convergence for (x 2) n n=4 3n 4. Find the interval convergence for x n n n 3 n St. Francis High School Page 12 12 AP Calc BC