In this section, we show how to use the integral test to decide whether a series

Transcription

1 Itegral Test Itegral Test Example Itegral Test Example p-series Compariso Test Example Example 2 Example 3 Example 4 Example 5 Exa Itegral Test I this sectio, we show how to use the itegral test to decide whether a series X of the form (where a ) coverges or diverges by comparig it to a p =a improper itegral. Itegral Test Suppose f (x) is a positive decreasig cotiuous fuctio o the iterval [, ) with f () = a. The the series P = a is coverget if ad oly if R f (x)dx coverges, that is: Z X If f (x)dx is coverget, the is coverget. If Z f (x)dx is diverget, the = X = a a is diverget. Note The result is still true if the coditio that f (x) is decreasig o the iterval [, ) is relaxed to the fuctio f (x) is decreasig o a iterval [M, ) for some umber M.

2 Itegral Test Itegral Test Example Itegral Test Example p-series Compariso Test Example Example 2 Example 3 Example 4 Example 5 Exa Itegral Test (Why it works: covergece) We kow from a previous lecture that dx coverges if p > ad diverges if p. x p R I the picture we compare the series P = itegral R to the improper 2 dx. x 2 The k th partial sum is s k = + P k =2 < + R 2 x 2 dx = + = 2. Sice the sequece {s k } is icreasig (because each a > 0) ad bouded, we ca coclude that the sequece of partial sums coverges ad hece the series X coverges. 2 i= NOTE We are ot sayig that P i= = R 2 x 2 dx here.

3 Itegral Test Itegral Test Example Itegral Test Example p-series Compariso Test Example Example 2 Example 3 Example 4 Example 5 Exa Itegral Test (Why it works: divergece) We kow that R x p dx coverges if p > ad diverges if p. I the picture, we compare the series P = to the improper itegral R x dx. X = k= 2 3 This time we draw the rectagles so that we get s > s = + + Z + + > dx 2 3 x Thus we see that lim s > lim R However, we kow that R R x dx. x dx diverges, we ca coclude that P k= x dx grows without boud ad hece sice also diverges.

4 Itegral Test Itegral Test Example Itegral Test Example p-series Compariso Test Example Example 2 Example 3 Example 4 Example 5 Exa p-series We kow that R X = x p dx coverges if p > ad diverges if p. p coverges for p >, diverges for p. Example Determie if the followig series coverge or diverge: X = 3, X = 5, X =0 5, X =00 5, P = P P = =0 5 3 diverges sice p = /3 <. coverges sice p = 5 >. 5 also coverges sice a fiite umber of terms have o effect whether a series coverges or diverges. P =00 5 cov/diverges if ad oly if P = sice p = /5 <. 5 cov/div. This diverges

5 Itegral Test Itegral Test Example Itegral Test Example p-series Compariso Test Example Example 2 Example 3 Example 4 Example 5 Exa Compariso Test I this sectio, as we did with improper itegrals, we see how to compare a series (with Positive terms) to a well kow series to determie if it coverges or diverges. We will of course make use of our kowledge of p-series ad geometric series. X = p coverges for p >, diverges for p. X ar coverges if r <, diverges if r. = Compariso Test Suppose that P a ad P b are series with positive terms. (i) If P b is coverget ad a b for all, tha P a is also coverget. (ii) If P b is diverget ad a b for all, the P a is diverget.

6 Itegral Test Itegral Test Example Itegral Test Example p-series Compariso Test Example Example 2 Example 3 Example 4 Example 5 Exa Example Example Use the compariso test to determie if the followig series coverges or diverges: X 2 / 3 = First we check that a > 0 > true sice 2 / 3 > 0 for. We have 2 / = 2 > for. Therefore 2 / = < for. 2 Therefore 2 / 3 < 3 for >. Sice P = 3 is a p-series with p >, it coverges. Comparig the above series with P P = 2 / 3 = also coverges ad P =, we ca coclude that 3 P = 2 / 3 3

7 Itegral Test Itegral Test Example Itegral Test Example p-series Compariso Test Example Example 2 Example 3 Example 4 Example 5 Exa Example 2 Example 2 Use the compariso test to determie if the followig series coverges or diverges: X 2 / First we check that a > 0 > true sice 2/ > 0 for. = We have 2 / = 2 > for. Therefore 2/ Sice P = diverges. > for >. is a p-series with p = (a.k.a. the harmoic series), it Therefore, by compariso, we ca coclude that P 2 / = also diverges.

8 Itegral Test Itegral Test Example Itegral Test Example p-series Compariso Test Example Example 2 Example 3 Example 4 Example 5 Exa Example 3 Example 3 Use the compariso test to determie if the followig series coverges or diverges: X 2 + = First we check that a > 0 > true sice We have 2 + > 2 for. Therefore Sice P = < for > > 0 for. 2 + is a p-series with p = 2, it coverges. Therefore, by compariso, we ca coclude that P ad P = P 2 + = 2. = also coverges 2 +

9 Itegral Test Itegral Test Example Itegral Test Example p-series Compariso Test Example Example 2 Example 3 Example 4 Example 5 Exa Example 4 Example 4 Use the compariso test to determie if the followig series coverges or diverges: X 2 2 = First we check that a > 0 > true sice 2 = > 0 for We have < for Sice P = 2 is a p-series with p = 2, it coverges. Therefore, by compariso, we ca coclude that P ad P = P 2 2 =. 2 = 2 2 also coverges

10 Itegral Test Itegral Test Example Itegral Test Example p-series Compariso Test Example Example 2 Example 3 Example 4 Example 5 Exa Example 5 Example 5 Use the compariso test to determie if the followig series coverges or diverges: X l = First we check that a > 0 > true sice l > > 0 for e. Note that this allows us to use the test sice a fiite umber of terms have o bearig o covergece or divergece. We have l Sice P = > for > 3. diverges, we ca coclude that P = l also diverges.

11 Itegral Test Itegral Test Example Itegral Test Example p-series Compariso Test Example Example 2 Example 3 Example 4 Example 5 Exa Example 6 Example 6 Use the compariso test to determie if the followig series coverges or diverges: X! = First we check that a > 0 > true sice > 0 for.! We have! = ( )( 2) 2 > = 2. Therefore <.! 2 Sice P = coverges, we ca coclude that P 2 = also coverges.!

12 Itegral Test Itegral Test Example Itegral Test Example p-series Compariso Test Example Example 2 Example 3 Example 4 Example 5 Exa Limit Compariso Test Limit Compariso Test Suppose that P a ad P b are series with positive terms. If a lim = c b where c is a fiite umber ad c > 0, the either both series coverge or both diverge. (Note c 0 or. ) Example Test the followig series for covergece usig the Limit Compariso test: X 2 =2 (Note that our previous compariso test is difficult to apply i this ad most of the examples below.) First we check that a > 0 > true sice a = > 0 for 2. (after 2 we study absolute covergece, we see how to get aroud this restrictio.) We will compare this series to P =2 p-series with p = 2. b = 2. 2 which coverges, sice it is a lim a b = lim /( 2 ) / 2 = lim 2 2 = lim (/ 2 ) = Sice c = > 0, we ca coclude that both series coverge.

13 Itegral Test Itegral Test Example Itegral Test Example p-series Compariso Test Example Example 2 Example 3 Example 4 Example 5 Exa Example Limit Compariso Test Suppose that P a ad P b are series with a positive terms. If lim b = c where c is a fiite umber ad c > 0, the either both series coverge or both diverge. (Note c 0 or. ) Example Test the followig series for covergece usig the Limit Compariso test: X = First we check that a > 0 > true sice a = > 0 for For a ratioal fuctio, the rule of thumb is to compare the series to the series P p, where p is the degree of the umerator ad q is the degree of q the deomiator. We will compare this series to P 2 = 4 it is a p-series with p = 2. b = 2. = P = 2 which coverges, sice lim a b = lim ( )/(/2 ) = lim = lim +2/+/ 2 +/ 2 +2/ 3 +/ 4 =. Sice c = > 0, we ca coclude that both series coverge.

14 Itegral Test Itegral Test Example Itegral Test Example p-series Compariso Test Example Example 2 Example 3 Example 4 Example 5 Exa Example Example Test the followig series for covergece usig the Limit Compariso test: X = First we check that a > 0 > true sice a = 2+ > 0 for. 3 + We will compare this series to P = = P 3 = = P 3/2 = diverges, sice it is a p-series with p = /2. b =. which. a lim b = lim ( 2+ ) (/ +2 ) = lim 3/2 + = /2 (2+/) = lim (+/ = 2. 3/2 3 ) lim (2 3/2 + ) 3 + (2+/) = ( 3 lim +)/3 Sice c = 2 > 0, we ca coclude that both series diverge.

15 Itegral Test Itegral Test Example Itegral Test Example p-series Compariso Test Example Example 2 Example 3 Example 4 Example 5 Exa Example Example Test the followig series for covergece usig the Limit Compariso test: X e 2 = First we check that a > 0 > true sice a = e > 0 for. 2 We will compare this series to P = which coverges, sice it is a 2 geometric series with r = /2 <. b = 2. lim a b. e = lim ( ) (/2 e ) = lim 2 = e. /2 Sice c = e > 0, we ca coclude that both series coverge.

16 Itegral Test Itegral Test Example Itegral Test Example p-series Compariso Test Example Example 2 Example 3 Example 4 Example 5 Exa Example Example Test the followig series for covergece usig the Limit Compariso test: X 2 / 2 = First we check that a > 0 > true sice a = 2/ 2 > 0 for. We will compare this series to P = p-series with p = 2 >. b = 2. 2 which coverges, sice it is a. a lim b = lim ( 2/ ) (/ 2 ) = lim 2 2 / = lim e l 2 =. Sice c = > 0, we ca coclude that both series coverge.

17 Itegral Test Itegral Test Example Itegral Test Example p-series Compariso Test Example Example 2 Example 3 Example 4 Example 5 Exa Example Example Test the followig series for covergece usig the Limit Compariso test:! X = First we check that a > 0 > true sice a = +! 3 3 > 0 for. We will compare this series to P = which coverges, sice it is a 3 geometric series with r = /3 <. b = 3.! 3!. 3 a lim b = lim ( + 3 ) (/3 ) = lim + =. Sice c = > 0, we ca coclude that both series coverge.

18 Itegral Test Itegral Test Example Itegral Test Example p-series Compariso Test Example Example 2 Example 3 Example 4 Example 5 Exa Example Example Test the followig series for covergece usig the Limit Compariso test: X π si = First we check that a > 0 > true sice a = si π > 0 for >. We will compare this series to P π = = π P = which diverges, sice it is a costat times a p-series with p =. b = π. a lim b = lim (si π. ) ( π si x ) = limx 0 =. x Sice c = > 0, we ca coclude that both series diverge.