Section 10.3 Convergence of series with positive terms 1. Integral test 2. Error estimates for the integral test 3. Comparison test 4. Limit comparison test (LCT) Math 126 Enhanced 10.3 Series with positive terms The University of Kansas 1 / 12
Remember: If a n converges, then lim n a n = 0. Test for divergence is equivalent to the If lim n a n 0, then a n diverges. BUT lim n a n = 0 does not imply a n converges. Math 126 Enhanced 10.3 Series with positive terms The University of Kansas 2 / 12
iclicker Question 1 If {a n } is a convergent sequence, then a n is a convergent series. (A) True (B) False Math 126 Enhanced 10.3 Series with positive terms The University of Kansas 3 / 12
Integral Test for Convergence Given a series a n such that a n 0 and a continuous, positive, decreasing function f (x) such that f (n) = a n for every n, then a n converges if and only if 1 f (x) dx converges. Math 126 Enhanced 10.3 Series with positive terms The University of Kansas 4 / 12
Integral Test for Convergence Given a series a n such that a n 0 and a continuous, positive, decreasing function f (x) such that f (n) = a n for every n, then a n converges if and only if 1 f (x) dx converges. Math 126 Enhanced 10.3 Series with positive terms The University of Kansas 4 / 12
Question When does 1 n p converge? Math 126 Enhanced 10.3 Series with positive terms The University of Kansas 5 / 12
Question When does 1 n p converge? P-Series test 1 converges if and only if p > 1. np Math 126 Enhanced 10.3 Series with positive terms The University of Kansas 5 / 12
Exercise Determine if the following series converge or diverge. 3 1. n 4 Math 126 Enhanced 10.3 Series with positive terms The University of Kansas 6 / 12
Exercise Determine if the following series converge or diverge. 3 1. n 4 1 2. 3 n n=2 Math 126 Enhanced 10.3 Series with positive terms The University of Kansas 6 / 12
Exercise Determine if the following series converge or diverge. 3 1. 3. ne n2 n 4 n=3 1 2. 3 n n=2 Math 126 Enhanced 10.3 Series with positive terms The University of Kansas 6 / 12
Exercise Determine if the following series converge or diverge. 3 1. 3. ne n2 n 4 n=3 1 n 2. 3 4. n 10n + 12 n=2 n=4 Math 126 Enhanced 10.3 Series with positive terms The University of Kansas 6 / 12
iclicker Question 2 The series n=2 1 n ln n (A) Converges (B) Diverges (C) Not enough information Math 126 Enhanced 10.3 Series with positive terms The University of Kansas 7 / 12
Remainder Estimate for the Integral Test error term S s k k f (x) dx Math 126 Enhanced 10.3 Series with positive terms The University of Kansas 8 / 12
Remainder Estimate for the Integral Test error term S s k k f (x) dx Exercise Estimate the following sums. First show that the integral test is valid for each sum. 1 1. n 3 to within 10 2 Math 126 Enhanced 10.3 Series with positive terms The University of Kansas 8 / 12
Remainder Estimate for the Integral Test error term S s k k f (x) dx Exercise Estimate the following sums. First show that the integral test is valid for each sum. 1 1. 2. (2n + 1) 6 n 3 to within 10 2 to four decimal places Math 126 Enhanced 10.3 Series with positive terms The University of Kansas 8 / 12
The comparison test Suppose that {a n } and {b n } are positive sequences whose terms satisfy 0 b n a n. Math 126 Enhanced 10.3 Series with positive terms The University of Kansas 9 / 12
The comparison test Suppose that {a n } and {b n } are positive sequences whose terms satisfy 0 b n a n. (i) If a n converges then b n converges. (ii) If b n diverges then a n diverges. Math 126 Enhanced 10.3 Series with positive terms The University of Kansas 9 / 12
The comparison test Suppose that {a n } and {b n } are positive sequences whose terms satisfy 0 b n a n. (i) If a n converges then b n converges. (ii) If b n diverges then a n diverges. Exercise Determine if the following series converge or diverge. 1 1. 3 n + n n=0 Math 126 Enhanced 10.3 Series with positive terms The University of Kansas 9 / 12
The comparison test Suppose that {a n } and {b n } are positive sequences whose terms satisfy 0 b n a n. (i) If a n converges then b n converges. (ii) If b n diverges then a n diverges. Exercise Determine if the following series converge or diverge. 1 ln(n) 1. 2. 3 n + n n n=0 n=3 Math 126 Enhanced 10.3 Series with positive terms The University of Kansas 9 / 12
If you want to use the comparison test, but it is unclear what comparison you need to make, try: The limit comparison test Given a series a n, choose a series b n that is similar but easier to determine convergence. To apply the LCT, we need a n, b n > 0. Let a n L = lim. n b n 1. If 0 < L < (finite but not 0) then whatever happens to bn happens to a n as well. 2. If L = and b n diverges, then a n diverges. 3. If L = 0 and b n converges, then a n converges. Math 126 Enhanced 10.3 Series with positive terms The University of Kansas 10 / 12
Exercise Determine if the following series converge or diverge. r 2 1. r 5 + r 3 + 1 r=1 Math 126 Enhanced 10.3 Series with positive terms The University of Kansas 11 / 12
Exercise Determine if the following series converge or diverge. r 2 1. r 5 + r 3 + 1 2. r=1 m=1 2 m + 4 m 3 m + 5 m Math 126 Enhanced 10.3 Series with positive terms The University of Kansas 11 / 12
Exercise Determine if the following series converge or diverge. r 2 n 1 1. 3. r 5 + r 3 + 1 2n 2 + 3 2. r=1 m=1 2 m + 4 m 3 m + 5 m n=3 Math 126 Enhanced 10.3 Series with positive terms The University of Kansas 11 / 12
Exercise Determine if the following series converge or diverge. r 2 n 1 1. 3. r=1 r 5 + r 3 + 1 2n 2 + 3 n=3 2 m + 4 m ( ) 1 2. 4. sin 3 m + 5 m n 1.3 m=1 Math 126 Enhanced 10.3 Series with positive terms The University of Kansas 11 / 12
iclicker Question 3 Determine the convergence of 1 n(ln n ) 2 n=3 (A) Diverges by the Integral Test. (B) Diverges by the Comparison Test in comparison to 1 n. (C) Convergence by the Test for Divergence. (D) Convergence by the Comparison Test in comparison to 1 n 3. (E) Convergence by the Integral Test. Math 126 Enhanced 10.3 Series with positive terms The University of Kansas 12 / 12