The Comparison Test. J. Gonzalez-Zugasti, University of Massachusetts - Lowell

Transcription

1 The Comparison Test

2 The Comparison Test Let a k and b k be series with positive terms and suppose a N b N, a N+ b N+, a N+2 b N+2,, a) If the bigger series b k converges, then the smaller series a k also converges. b) On the other hand, if the smaller series a k diverges, then the bigger series b k also diverges. 2

3 Informal Principle # Suppose u k is series with positive terms. Constant terms in the denominator of u k can usually be deleted without affecting the convergence of divergence of the series. 3

4 Example Guess if the series converge or diverge. a) b) c) k= k= k= 2 k + k 2 k

5 Example (continued) Solution: (a) k= we expect to behave like k= 2 k + 2 k, which is a convergent geometric series (a = 2, r = 2 ) 5

6 Example (continued) (b) k= we expect to behave like k 2 k=, which is a divergent p-series (p = ) 2 k (c) k= 3 we expect to behave like k+ 2 k=, which is a convergent p-series k 3 (p = 3) 6

7 Informal Principle #2 If a polynomial in k appears as a factor in the numerator or denominator of u k, all but the highest power of k in the polynomial may usually be deleted without affecting the convergence of divergence behavior of the series. 7

8 Example 2 Guess if the series converge or diverge. a) b) k= k= k 3 +2k 6k 4 2k 3 + k 5 +k 2 2k 8

9 Example 2 (continued) Solution: (a) k= we expect to behave like k 3 +2k k=, which is a convergent p-series (p = 3 ). 2 k 3 6k 4 2k 3 + (b) k= we expect to behave like k= k 5 +k 2 2k 6k 4 k 5 = 6 k= the divergent harmonic series. k, which is a constant times 9

10 Example 3 Use the Comparison Test to determine whether converges or diverges. Solution: 2k 2 + k k= We expect this series to behave like 2 k= (p = 2). k 2 k= = 2k 2, which is a constant times a convergent p-series 0

11 Example 3 (continued) Since we expect the series to converge, we want to find b k such that b k converges and 2k 2 + k b k. Notice 2k 2 + k for k. 2k2 Since k= converges, so does 2k 2 the Comparison Test. k= by 2k 2 +k

12 Example 4 Use the Comparison Test to determine whether converges or diverges. Solution: 2k 2 k k= We expect this series to behave like 2 k= (p = 2). k 2 k= = 2k 2, which is a constant times a convergent p-series 2

13 Example 4 (continued) Since we expect the series to converge, we want to find b k such that b k converges and 2k 2 k b k. Unfortunately 2k 2 k > for k. 2k2 2k 2 So k= cannot be our choice for b k. 3

14 Example 4 (continued) We want to decrease the denominator of k 2 2k 2 k. If we try b k = = 2k 2 k 2 k2 we get 2k 2 k 2k 2 k 2 = for k. k2 Since k= is a convergent p-series (p = 2), our series k= converges by the 2k 2 k Comparison Test. 4

15 Example 5 Use the Comparison Test to determine whether converges or diverges. k= k 4 Solution: We expect this series to behave like the divergent harmonic series. k= k, which is 5

16 Example 5 (continued) Since we expect the series to diverge, we want to find a k such that a k diverges and a k k. 4 Notice k k for k. 4 Since k= diverges, our series k k= k diverges by the 4 Comparison Test. 6

17 Example 6 Use the Comparison Test to determine whether converges or diverges. k + 5 k= Solution: We expect this series to behave like divergent p-series (p = ). 2 k= k, which is a 7

18 Example 6 (continued) Since we expect the series to diverge, we want to find a k such that a k diverges and a k k + 5. Unfortunately, k for k. k + 5 So k= cannot be our choice for a k. k 8

19 Example 6 (continued) We want to increase the denominator of k+5. If we try a k = = we get k+ k 2 k k + k = 2 k for k 25. k + 5 Since k= = k 2 k 2 k= is a constant times a divergent p-series (p = ), 2 our series k= diverges by the Comparison k+5 Test. 9

20 The Limit Comparison Test Let a k and b k be series with positive terms and suppose a k ρ = lim k b k a) If ρ is finite and ρ > 0, then the series both converge or both diverge. b) If ρ = 0 and b k converges, then a k converges. c) If ρ = and b k diverges, then a k diverges. 20

21 Example 7 Use the Limit Comparison Test to determine whether 3k3 2k k 5 k k= converges or diverges. Solution: 3k 3 We expect this series to behave like k= = k 5 k= = k 2 3 k=, which is a constant times a convergent p-series k (p = 2)

22 Example 7 (continued) ρ = lim k 3k 3 2k k 5 k k 2 3k 3 2k = lim k k 5 k k2 3 Since this is a rational function with the degree of the numerator equal to the degree of the denominator (=5), this limit is equal to the ratio of the leading coefficients. 22

23 Example 7 (continued) ρ = lim k 3k 3 2k k 5 k k2 3 = 3 3 = > 0 So, by the Limit Comparison Test, 3k 3 2k 2 +4 k= converges. k 5 k

24 Thomas Simpson (720-76) Simpson was a successful text writer and did most of his work in probability. He taught at the Royal Military Academy in Woolwich. His first articles were published in the Ladies' Diary. Later he became editor of this popular journal. Simpson's rule to approximate definite integrals was developed and used before he was born. It is another of history's beautiful quirks that one of the ablest mathematicians of the 8th century is remembered not for his own work or his textbooks but for a rule that was never his, that he never claimed, and that bears his name only because he happened to mention it in one of his books. 24