10.4 Comparison Tests

Transcription

1 0.4 Comparison Tests

2 The Statement Theorem Let a n be a series with no negative terms. (a) a n converges if there is a convergent series c n with a n c n n > N, N Z (b) a n diverges if there is a divergent series of nonnegative terms d n with a n d n n > N, N Z

3 The Statement Theorem Let a n be a series with no negative terms. (a) a n converges if there is a convergent series c n with a n c n n > N, N Z (b) a n diverges if there is a divergent series of nonnegative terms d n with a n d n n > N, N Z The idea is that for some very difficult looking series, we can use either known or easier series to determine convergence. This is similar to the Comparison Test for integrals that we talked about in chapter 8.

4 First Determine if the following series converges: k= k 2

5 First Determine if the following series converges: k= k 2 What is a good choice for comparison?

6 First Determine if the following series converges: k= k 2 What is a good choice for comparison? We know that diverges because p = 2 <. k= k

7 First Determine if the following series converges: k= k 2 What is a good choice for comparison? We know that k= k diverges because p = 2 <. Since k > for all k 2 k =, 2,, we have, by the Comparison Test, that our series diverges as well.

8 Second Exmaple Determine if the following series converges: k= 2k 2 + k

9 Second Exmaple Determine if the following series converges: k= 2k 2 + k Good choice for comparison?

10 Second Exmaple Determine if the following series converges: k= 2k 2 + k Good choice for comparison? We know that k= 2k 2 = 2 k= converges by the P-series test and multiplication by a constant does not change convergence - only the value of the sum and if we are only concerned about convergence, we don t care about the sum right now. Also, we know 2k 2 +k < for k =, 2,, 2k 2 so our series converges as well. k 2

11 The Limit Comparison Test Theorem Suppose that a n > 0 and b n > 0 for all n N, N Z. a (a) If lim n n b n (b) If lim n (c) If lim n a n b n a n b n = c > 0, then a n and b n both converge or diverge. = 0 and b n converges, then a n converges. = and b n diverges then a n diverges.

12 The Limit Comparison Test Theorem Suppose that a n > 0 and b n > 0 for all n N, N Z. a (a) If lim n n b n (b) If lim n (c) If lim n a n b n a n b n = c > 0, then a n and b n both converge or diverge. = 0 and b n converges, then a n converges. = and b n diverges then a n diverges. This is very much like the related rates (rates of growth) section we did (section 7.4).

13 First Determine if the given series converges. k= k

14 First Determine if the given series converges. k= k This one could be done using the comparison test as well, and we will use the same series for comparison that we would if we were using that test as well. When using this test, we first want to decide whether we think the series converges or diverges and we want to choose a known series that behaves similarly.

15 First Determine if the given series converges. k= k This one could be done using the comparison test as well, and we will use the same series for comparison that we would if we were using that test as well. When using this test, we first want to decide whether we think the series converges or diverges and we want to choose a known series that behaves similarly. What would be a good choice here?

16 First Here we choose k, which we have seen diverges by the P-series test.

17 First Here we choose k, which we have seen diverges by the P-series test. Then, since lim k a k b k

18 First Here we choose k, which we have seen diverges by the P-series test. Then, since = lim k a lim k k b k k k

19 First Here we choose k, which we have seen diverges by the P-series test. Then, since = lim k = lim k a lim k k b k k k k k =

20 First Here we choose k, which we have seen diverges by the P-series test. Then, since = lim k = lim k a lim k k b k k k k k = By the Limit Comparison Test (), since k diverges, so does our series.

21 Second Determine if the following series converges: k= 2k 2 + k

22 Second Determine if the following series converges: k= 2k 2 + k Choice?

23 Second Determine if the following series converges: k= 2k 2 + k Choice? We will use 2k 2 as we did using the Comparison Test.

24 Second Determine if the following series converges: k= 2k 2 + k Choice? We will use 2k 2 as we did using the Comparison Test. We know that 2k 2 converges, and since a lim k = lim k b k k 2k 2 +k 2k 2 = we can again invoke the Limit comparison Test to conclude that converges also. 2k 2 +k

25 Last Determine if the following series converges: 3k 3 2k k k= 7 k 3 + 2

26 Last Determine if the following series converges: 3k 3 2k k k= 7 k What does this expression behave like?

27 Last Determine if the following series converges: 3k 3 2k k k= 7 k What does this expression behave like? This rational expression behaves live 3 k 4, so we will use b k = 3 k 4. We know that 3 x 4 converges by the P-series test.

28 Last Now, lim k a k b k

29 Last Now, a lim k = lim k b k k 3k 3 2k 2 +4 k 7 k k 4

30 Last Now, a lim k = lim k b k k 3k 3 2k 2 +4 k 7 k k 4 3k = 3 2k lim k k 7 k k4 3

31 Last Now, a lim k = lim k b k k 3k 3 2k 2 +4 k 7 k k 4 3k = 3 2k lim k k 7 k k4 3 When we simplify this, we get 3k 7 2k 6 + 4k 4 lim k 3k 7 3k =

32 Last Now, a lim k = lim k b k k 3k 3 2k 2 +4 k 7 k k 4 3k = 3 2k lim k k 7 k k4 3 When we simplify this, we get 3k 7 2k 6 + 4k 4 lim k 3k 7 3k = So, since b k converges, we know our series does as well by the Limit Comparison Test.