Convergence Tests. Academic Resource Center

Transcription

1 Convergence Tests Academic Resource Center

2 Series Given a sequence {a 0, a, a 2,, a n } The sum of the series, S n = A series is convergent if, as n gets larger and larger, S n goes to some finite number. If S n does not converge, and S n goes to, then the series is said to be divergent n k a k

3 Geometric and P-Series The two series that are the easiest to test are geometric series and p-series. Geometric is generally in the form P-series is generally in the form k ar k n n p

4 Geometric Series A geometric series is a series in which there is a constant ratio between successive terms each successive term is the previous term multiplied by 2 each successive term is the previous term squared

5 Geometric Series S n = = ar ar +ar^2 + ar^3 + +ar^k k k As a result, if r <, the geometric series will converge to, and if r the series will diverge. a r

6 P-Series Given a series n n p This series is said to be convergent if p>, And divergent if p

7 Geometric and P-Series Examples 3 n n 4 n n So S= 3/(-3/4) = 2 n This series is geometric with a=3 and r =3/4. Since r<, this series will converge. The Sum of the series, S a S r n n 3 Here, p=3, so p>. Therefore our series will converge n n n n 2 Here, p=/2, so p<. Therefore our series will diverge

8 Convergence Tests Divergence test Comparison Test Limit Comparison Test Ratio Test Root Test Integral Test Alternating Series Test

9 Divergence Test Say you have some series The easiest way to see if a series diverges is this test Evaluate L= Lim If L 0, the series diverges If L=0, then this test is inconclusive n a n n 0 a n

10 Divergence Test Example n n 2 5 n 2 4 Let s look at the limit of the series Lim n n 2 5 n 2 4 Lim n n 2 5 n Therefore, this series is divergent n Lim n n 2 n 0 2 The limit here is equal to zero, so this test is inconclusive. However, we should see that this a p-series with p>, therefore this will converge.

11 Comparison Test Often easiest to compare geometric and p series. Let and be series with non-negative terms. If a k for k, as k gets big, then b k If converges, then converges If diverges, then diverges b k a k b k a k b k a k

12 Comparison Test Example n 3 n This is very similar to And since n 3 n Test to see if this series converges using the comparison test n n 3 n 3 n which is a geometric series so it will converge our original series will also converge

13 Limit Comparison Test Let and be series with non-negative terms. a k Evaluate Lim If lim=l, some finite number, a then both k either converge or diverge. b k and and are generally geometric series or p-series, so seeing whether these series are convergent is fast. b k k b k a k a k b k

14 Limit Comparison Test Example n 9 n 3 0 n Compare it with And since n Determine whether this series converges or not n 9 n 0 n n 9 0 n n 9 0 so Lim n Is a geometric series with r<, this series converges, therefore so does our original series 9 n 3 0 n 0 n 9 0

15 Ratio Test Let a k be a series with non-negative terms. Evaluate L= Lim If L <, then If L >, then converges diverges If L =, then this test is inconclusive k a k a k a k a k

16 Ratio Test Example Test for convergence n n 3 3 n n Look at the limit of a n a n Lim n () n (n ) 3 3 n () n n 3 3 n Lim n (n ) 3 3 n 3 n n 3 3 Lim ( n n n ) 3 3 Lim ( n n ) 3 3 Since L<, this series will converge based on the ratio test

17 Root Test Let a n Useful if Evaluate L= Lim If L <, If L >, be a series with non-negative terms. involves nth powers a n is convergent n is divergent If L =, then this test is inconclusive a n a n (a n ) n

18 Root Test Example Test for convergence n ( 4 n 5 5 n 6 ) n Lim (( 4 n 5 n 5 n 6 ) n ) Lets evaluate the limit, L =Lim n Lim n By the root test, since L<, our series will converge. n (a n ) 4 n 5 5 n n

19 Integral Test Given the series, let = f(k) a k f must be continuous, positive, and decreasing for x > 0 will converge only if converges. If diverges, then the series will also diverge. In general, however, k 0 a k 0 f ( x )dx 0 a k f ( x )dx a k f ( x k )dx

20 Integral Test Example Test for convergence n (2 n ) 3 So let f ( x ) (2 x ) 3 Since x>0, f(x) is continuous and positive. f (x) is negative so we know f(x) is decreasing. Now let s look at the integral (2 x ) 3 Lim t ( dx 2 2 Lim [ t (2 t ) 2 (3) ) 2 9 Note: Series will most likely not converge to /9, but it will converge nonetheless. ] (2 x ) 2 Since the integral converged to a finite number, our original series will also converge t

21 Alternating Series Test Given a series, where is positive for all k IF for all k, and Lim = 0 a k a k () k a k Then the series is convergent a k k a k

22 Alternating Series Test Example Test for convergence n n n 2 n 3 4 Check: Is this series decrease- yes Is the Lim=0? Lim n n 2 Therefore, n n Yes n n 2 n 3 4, is convergent.

23 More Examples. n cos n n n n n 2 n 2 n 6 3. n 0 n (n )4 2 n 4. n 2 n ln n

24 Answers. By Alternating series test, series will converge 2. By the comparison test, series will diverge 3. By the ratio test, series will converge 4. By the integral test, series will diverge