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
∞ ( X 1 converges for = np n=1 diverges for ∞ X Z ∞ 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, ∞). 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 ∞ ∞ X X 0 ≤ an ≤ bn n=1 n=1
P∞ P∞ n=1 an n=1 bn P∞ P∞ n=1 bn n=1 an
The (Basic) Comparison Test
Theorem (The (Basic) Comaprison Test)
Suppose {an} and {bn} 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→∞ bn where L is a finite number and L 6= 0.
∞ ∞ X X an converges if and only if bn converges. n=1 n=1
Theorem (The Limit Comaprison Test)
Suppose {an} and {bn} are two sequences such that
Then Example P∞ k sin2 k Determine the convergence of k=2 k3−1 Example P∞ log√n 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.
∞ ∞ X 1 X 1 2n 2n + 1 n=1 n=1 ∞ ∞ X 1 X 1 2n 2n − 1 n=1 n=1 ∞ ∞ X 1 X n2 + 1 n2 n4 − n2 + 1 n=1 n=1 ∞ ∞ √ 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.
∞ ∞ X sin n + 2 X 1 n2 n2 n=1 n=1 ∞ ∞ X n sin2 n + cos n + 3 X 1 n4 + 5 − 4 cos n n3 n=1 n=1 ∞ ∞ X X 12e−n e−n log(n + 1) n=1 n=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 P∞ (−1)n+3 Determine the convergence of n=1 n . Example P∞ (−1)n+53n2 Determine the convergence of n=1 4n2+1 .