Math 115 Calculus II 11.3 The Integral Test, 11.4 Comparison Tests, 11.5 Alternating Series Mikey Chow March 23, 2021 Table of Contents Integral Test Basic and Limit Comparison Tests Alternating Series p-series 1 ( X 1 converges for = np n=1 diverges for 1 X Z 1 an converges if and only if f (x)dx converges. n=1 1 The Integral Test Theorem (The Integral Test) Suppose the sequence an is given by f (n) where f is a continuous, positive, decreasing sequence on [1; 1). Then Example Determine the convergence of the series of the sequence e−1; 2e−2; 3e−3;::: Table of Contents Integral Test Basic and Limit Comparison Tests Alternating Series 1 1 X X 0 ≤ an ≤ bn n=1 n=1 P1 P1 n=1 an n=1 bn P1 P1 n=1 bn n=1 an The (Basic) Comparison Test Theorem (The (Basic) Comaprison Test) Suppose fang and fbng are two sequences such that 0 ≤ an ≤ bn for all n. Then In particular, If converges, then converges also. If diverges, then diverges also. (What does this remind you of?) Remark We can still apply the basic comparison test if we know that a lim n = L n!1 bn where L is a finite number and L 6= 0. 1 1 X X an converges if and only if bn converges. n=1 n=1 The Limit Comparison Test Theorem (The Limit Comaprison Test) Suppose fang and fbng are two sequences such that Then Example P1 k sin2 k Determine the convergence of k=2 k3−1 Example P1 logpn Determine the convergence of n=1 n n Group Problem Solving Instructions: For each pair of series below, determine the convergence of one of them without using a comparison test. Then use a comparison test with these two series to determine the convergence of the other. 1 1 X 1 X 1 2n 2n + 1 n=1 n=1 1 1 X 1 X 1 2n 2n − 1 n=1 n=1 1 1 X 1 X n2 + 1 n2 n4 − n2 + 1 n=1 n=1 1 1 p X 1 X i 2 + 1(i 3 + 3) n i 5 − i 4 + i 3 n=1 i=1 Group Problem Solving Instructions: For each pair of series below, determine the convergence of one of them without using a comparison test. Then use a comparison test with these two series to determine the convergence of the other. 1 1 X sin n + 2 X 1 n2 n2 n=1 n=1 1 1 X n sin2 n + cos n + 3 X 1 n4 + 5 − 4 cos n n3 n=1 n=1 1 1 X X 12e−n e−n log(n + 1) n=1 n=1 1 1 X e10−n X 1 n5 N5 n=1 N=1 Rough Work Rough Work Rough Work Table of Contents Integral Test Basic and Limit Comparison Tests Alternating Series stuff Alternating Sequences An alternating sequence is a sequence (an) of the form where Alternating Series Test Example P1 (−1)n+3 Determine the convergence of n=1 n . Example P1 (−1)n+53n2 Determine the convergence of n=1 4n2+1 ..