Section 8.3: The Integral and Comparison Tests; Estimating Sums

Transcription

1 Sectio 8.3: The Itegral ad Compariso Tests; Estimatig Sums Practice HW from Stewart Textbook (ot to had i) p. 585 # 3, 6-, 3-5 odd I this sectio, we wat to determie other methods for determiig whether a series coverges or diverges. The Itegral Test For a fuctio f, if f (x) > 0, is cotiuous ad decreasig for x M ad a = f (), the either both coverge or both diverge. a ad = M M f ( x) dx Note: The itegral test is oly a test for covergece or divergece. I the case of covergece, it does ot fid a value for the sum of the series. Example : Determie the covergece or divergece of the series = 4 + Solutio:

2 Example : Determie the covergece or divergece of the series = (l ) Solutio: We start by writig the formula for the sequece as a fuctio of x, that is, we write a = as f ( x) =. We should ote first of all that for x >, (l ) x (l x). f ( x) = is always positive (> 0),. cotiuous (the fuctio is oly udefied x (l x) whe x 0 ad whe x = sice l = 0), ad decreasig ), ad 3. decreasig (as x, f ( x) = 0 x (l x). The followig graph of this fuctio geerated usig Maple should help covice you of these facts: > f := x -> /(x*l(x)^); f := x x l() x > plot(f(x), x =..0, thickess =, view = [-..0, -..], title = "Graph of f(x) = /(x*l(x)^"); Thus, the itegral test ca be applied. We first set up the improper itegral of the fuctio ad itegrate as follows: (cotiued o ext page)

3 3 x (l x) dx = lim t t x (l x) dx x(l x) dx = Note to itegrate, use u du subs Let u = l x, du = dx x u du = = + = + = u du C C u u l x + C = lim t l x t (Result of itegral) = lim t l t l (Substitute the limits) = 0 + l (Evaluate the limit as t, l t 0) = l Sice the improper itegral evaluates to a fixed umber ( / l ), it is coverget. Thus by the itegral test, the series = (l ) is coverget.

4 4 Example 3: Show why the itegral test caot be used to aalyze the covergece or divergece of the series + ( ) Solutio: = p-series ad Harmoic Series A p-series series is give by = p p p p p = If p =, the 3 = = 4 + is called a harmoic series.

5 5 Covergece of p series A p-series = p p p p p = Coverges if p >.. Diverges if p. Example 4: Determie whether the p-series is coverget or diverget. Solutio: Example 5: Determie whether the p-series = is coverget or diverget. Solutio:

6 6 Example 6: Determie whether the p-series = Solutio: is coverget or diverget. Makig Comparisos betwee Series that are Similar May times we ca determie the covergece or divergece of a series by comparig it with the kow covergece or divergece of a related series. For example, = 3 + is close to the p-series =, 4 = 3 is close to the geometric series = 3 4. Uder the proper coditios, we ca use a series where it is easy to determie the covergece or divergece ad use it to determie covergece or divergece of a similar series usig types of compariso tests. We will examie two of these tests the direct compariso test ad the limit compariso test.

7 7 Suppose that a ad Direct Compariso Test b are series oly with positive terms ( a > 0 ad b > 0 ).. If b is coverget ad a b for all terms, a is coverget.. If b is diverget ad a b for all terms, a is diverget. Note: Most of the time, we will compare the give series to a p-series or a geometric series. Example 7: Determie whether the series is coverget or diverget. = 3 + Solutio:

8 8 Example 8: Determie whether the series = 4 3 is coverget or diverget. Solutio: Example 9: Demostrate why the direct compariso test caot be used to aalyze the covergece or divergece of the series ( ) = + Solutio: 0

9 9 Suppose that a ad ad Limit Compariso Test b are series oly with positive terms ( a > 0 ad b > 0 ) a lim = L where L is a fiite umber ad L > 0. b The either a ad b either both coverge or a ad b both diverge. Note: This test is useful whe comparig with a p-series. To get the p-series to compare with take the highest power of the umerator ad simplify. Example 0: Determie whether the series 5 3 = + 5 is coverget or diverget. Solutio: