The Comparison Test & Limit Comparison Test

Size: px

Start display at page:

Download "The Comparison Test & Limit Comparison Test"

Cuthbert Greene
5 years ago
Views:

1 The Comparison Test & Limit Comparison Test Math4 Department of Mathematics, University of Kentucky February 5, 207 Math4 Lecture 3 / 3

2 Summary of (some of) what we have learned about series... Math4 Lecture 3 2/ 3

3 Summary of (some of) what we have learned about series... Consider the series a n = a + a 2 + a 3 +. Math4 Lecture 3 2/ 3

4 Summary of (some of) what we have learned about series... Consider the series a n = a + a 2 + a 3 +. The terms a n form a sequence (a, a 2, a 3,...). Math4 Lecture 3 2/ 3

5 Summary of (some of) what we have learned about series... Consider the series a n = a + a 2 + a 3 +. The terms a n form a sequence (a, a 2, a 3,...). The n th partial sum is defined as S n = a + a a n = n i= a i. Math4 Lecture 3 2/ 3

6 Summary of (some of) what we have learned about series... Consider the series a n = a + a 2 + a 3 +. The terms a n form a sequence (a, a 2, a 3,...). The n th partial sum is defined as S n = a + a a n = n i= a i. The partial sums form a sequence: (S, S 2, S 3,...). Math4 Lecture 3 2/ 3

7 Summary of (some of) what we have learned about series... Consider the series a n = a + a 2 + a 3 +. The terms a n form a sequence (a, a 2, a 3,...). The n th partial sum is defined as S n = a + a a n = n i= a i. The partial sums form a sequence: (S, S 2, S 3,...). If the sequence of partial sums converges, so does a n. Furthermore, a n = lim n S n. Math4 Lecture 3 2/ 3

8 Summary of (some of) what we have learned about series... Consider the series a n = a + a 2 + a 3 +. The terms a n form a sequence (a, a 2, a 3,...). The n th partial sum is defined as S n = a + a a n = n i= a i. The partial sums form a sequence: (S, S 2, S 3,...). If the sequence of partial sums converges, so does a n. Furthermore, a n = lim n S n. If the sequence of partial sums diverges, so does a n. Math4 Lecture 3 2/ 3

9 Theorem If a n converges, then lim n a n = 0. Math4 Lecture 3 3/ 3

10 Theorem If a n converges, then lim n a n = 0. WARNING!! The converse is not true!! lim n a n = 0 = a n converges. Math4 Lecture 3 3/ 3

11 Theorem If a n converges, then lim n a n = 0. WARNING!! The converse is not true!! lim n a n = 0 = a n converges. All red cars are cars. BUT, not all cars are red. Math4 Lecture 3 3/ 3

12 Theorem If a n converges, then lim n a n = 0. WARNING!! The converse is not true!! lim n a n = 0 = a n converges. All red cars are cars. BUT, not all cars are red. Similarly, All convergent series a n possess the property that lim n a n = 0. BUT, not all series a n possessing the property that lim n a n = 0 are convergent. Math4 Lecture 3 3/ 3

13 Theorem If a n converges, then lim n a n = 0. WARNING!! The converse is not true!! lim n a n = 0 = a n converges. All red cars are cars. BUT, not all cars are red. Similarly, All convergent series a n possess the property that lim n a n = 0. BUT, not all series a n possessing the property that lim n a n = 0 are convergent. Test for Divergence If lim n a n does not exist, or if lim n a n 0, then a n diverges. Math4 Lecture 3 3/ 3

14 Other ways to test for convergence or divergence of a series? Math4 Lecture 3 4/ 3

15 Other ways to test for convergence or divergence of a series? Integral Test Suppose f is a continuous, positive, decreasing function on [, ) and let a n = f (n). Then, the series a n is convergent if and only if the improper integral f (x) dx is convergent. Math4 Lecture 3 4/ 3

16 Other ways to test for convergence or divergence of a series? Integral Test Suppose f is a continuous, positive, decreasing function on [, ) and let a n = f (n). Then, the series a n is convergent if and only if the improper integral f (x) dx is convergent. How can we determine whether Other tools?? 2 n + converges or diverges? Math4 Lecture 3 4/ 3

17 Example Determine whether the series divergent. 2 n + is convergent or Math4 Lecture 3 5/ 3

18 Example Determine whether the series divergent. Recall: 2 n = 2 n + }{{} 2 }{{} 2 a r is convergent or n = a r = /2 /2 =. Math4 Lecture 3 5/ 3

19 Example Determine whether the series divergent. Recall: 2 n = 2 }{{} a Note: 2 n + < 2 n for n =, 2,... 2 }{{} r 2 n + is convergent or n = a r = /2 /2 =. Math4 Lecture 3 5/ 3

20 Example Determine whether the series divergent. Recall: 2 n = 2 }{{} a Note: 2 n + < 2 n for n =, 2,... 2 }{{} r 2 n + is convergent or n = a r = /2 /2 = b n =/2 n a n =/(2 n +) n Math4 Lecture 3 5/ 3

21 Example Determine whether the series n is convergent or divergent. Math4 Lecture 3 6/ 3

22 Comparison Test (CT) Consider the series a n and b n, where a n, b n > 0 for every n =, 2,.... Then, (i) If b n is convergent and a n b n for every n, then a n is convergent as well. (ii) If b n is divergent and a n b n for every n, then a n is also divergent. Math4 Lecture 3 7/ 3

23 Comparison Test (CT) Consider the series a n and b n, where a n, b n > 0 for every n =, 2,.... Then, (i) If b n is convergent and a n b n for every n, then a n is convergent as well. (ii) If b n is divergent and a n b n for every n, then a n is also divergent. Key examples of series to keep in mind: () p-series n p (2) Geometric Series ar n Math4 Lecture 3 7/ 3

24 Comparison Test (CT) Consider the series a n and b n, where a n, b n > 0 for every n =, 2,.... Then, (i) If b n is convergent and a n b n for every n, then a n is convergent as well. (ii) If b n is divergent and a n b n for every n, then a n is also divergent. Key examples of series to keep in mind: () p-series n p (2) Geometric Series ar n Examples Using the Comparison Test, determine whether the following series converge or diverge. 3 n 2 n Math4 Lecture 3 7/ 3

25 Comparison Test (CT) Consider the series a n and b n, where a n, b n > 0 for every n =, 2,.... Then, (i) If b n is convergent and a n b n for every n, then a n is convergent as well. (ii) If b n is divergent and a n b n for every n, then a n is also divergent. Key examples of series to keep in mind: () p-series n p (2) Geometric Series ar n Examples Using the Comparison Test, determine whether the following series converge or diverge. 2 3 n 2 n 2 n 3 n Math4 Lecture 3 7/ 3

26 Example Determine whether the series 2 n converges or diverges. Comparison Test can t help us here because 2 n > 2, and n converges. Math4 Lecture 3 8/ 3

27 Example Determine whether the series 2 n converges or diverges. Comparison Test can t help us here because 2 n > 2, and n converges. Another approach: Limit Comparison Test Suppose a n and b n are series with a n, b n > 0. If a n lim = c, n b n where c is a positive constant, then either both series converge, or both series diverge. Math4 Lecture 3 8/ 3

28 Limit Comparison Test Suppose a n and b n are series with a n, b n > 0. If a n lim = c, n b n where c is a positive constant, then either both series converge, or both series diverge. c Math4 Lecture 3 9/ 3

29 Limit Comparison Test Suppose a n and b n are series with a n, b n > 0. If a n lim = c, n b n where c is a positive constant, then either both series converge, or both series diverge. M c m Math4 Lecture 3 0/ 3

30 Limit Comparison Test Suppose a n and b n are series with a n, b n > 0. If a n lim = c, n b n where c is a positive constant, then either both series converge, or both series diverge. M c m Math4 Lecture 3 / 3

31 Limit Comparison Test Suppose a n and b n are series with a n, b n > 0. If a n lim = c, n b n where c is a positive constant, then either both series converge, or both series diverge. M c m Math4 Lecture 3 2/ 3

32 Examples Use the Limit Comparison Test to determine whether the series below converge or diverge? 4 2 n+2 Math4 Lecture 3 3/ 3

33 Examples Use the Limit Comparison Test to determine whether the series below converge or diverge? 4 2 n n 2 + Math4 Lecture 3 3/ 3

The integral test and estimates of sums

The integral test and estimates of sums The integral test Suppose f is a continuous, positive, decreasing function on [, ) and let a n = f (n). Then the series n= a n is convergent if and only if the improper integral f (x)dx is convergent.