First, it is important to recall some key definitions and notations before discussing the individual series tests.

Transcription

1 Ifiite Series Tests Coverget or Diverget? First, it is importat to recall some key defiitios ad otatios before discussig the idividual series tests. sequece - a fuctio that takes o real umber values a ad is defied o the set of positive itegers, 2,,,... Listed Notatio - a { } a,a 2,a,K, a,k where for example series - a summatio of a sequece of umbers Notatio - a +, 2,,,... a a + a 2 + a +K+a +K covergece - wheever a sequece or a series has a limit divergece - wheever a sequece or a series does ot have a limit GEOMETRIC SERIES The first type of series test is used oly for a geometric series. A geometric series is a series which follows the patter, a + ar + ar 2 +K+ar +K where a is the iitial term ad r is a ratio (i.e., 2,, 5, etc. ). If r < (that is, < r < ), the the series coverges. a further, this series coverges to r Determie whether the series K coverges. Solutio: To fid r divide ay term by the term precedig it. The series follows the patter of a geometric series with a 2 ad r 2. By the test for covergece, this series does coverge, sice It follows that the series coverges to sice a r <.

2 POSITIVE SERIES A ifiite series with o egative terms is ofte referred to as a positive series. Positive series are usually deoted, a + a 2 + a +K+a +K There are several tests used to determie whether or ot a positive series coverges or diverges. THE DIVERGENCE TEST (Sectio.2): If lim a # 0, the the series diverges. Otherwise, the test is icoclusive. This test is useful i quickly determiig whether a give series diverges. It asks the questio, Is the series gettig cotiually larger? If it cotiues to grow, the the series diverges. Does the followig series coverge or diverge: Solutio: Examie the first couple of terms. Does the series cotiue to grow? The series is gettig cotiually larger. Thus lim K a # 0 ad the series diverges. THE INTEGRAL TEST (Sectio.): Let f (x) > 0 for x, ad f (x) is a cotiuous decreasig fuctio. Give a f (), if f (x)dx is coverget, the so is a ; if f (x)dx is diverget, the so is a. This test is useful for the iverse trig fuctios, as well as determiig the covergece or divergece of a particularly importat series, the harmoic series, which is the series i the example below. Determie whether or ot the series coverges K+ +K Solutio: The th-term test does t help here sice lim a 0 (the terms are gettig smaller). Now f (x) x is a cotiuous, positive, decreasig fuctio. Usig the itegral test b dx lim dx x x $ $ b lim(l b # l) b lim l x b # 0 Sice the itegral is diverget, the the series is diverget. Adapted from 2 b

3 P-SERIES TEST (Sectio.): If 0 < p, the the series If p >, the the series diverges. p coverges. p This test is derived from the itegral test ad is useful oly for the series which cotais the p term. However, it will be quite helpful i followig compariso tests. COMPARISON TEST (Sectio.): Give three series, p, d, c : if 0 p c for each ad c coverges, the p d coverges; if 0 d p for each ad diverges, the p diverges. This test is used quite ofte ad is oe of the more importat oes to kow. It requires familiarity with the covergece or divergece of other series, i additio to beig able to compare them with a give series. This test proves very useful whe a series appears u-testable. Example #: Does the series + 2 # + coverge or diverge? Solutio: This series closely resembles the p-series,, which coverges. I order to compare the two series usig the compariso test, determie which series is larger. Sice < for ay, + + <. (Multiplyig each side by + 2 # + coverges. preserves the iequality sice by the compariso test, is a positive term for all greater tha ) So Example #2: Does the series coverge or diverge? Solutio: Factor the deomiator, so the series ca be represeted as + 5 ( + ) Adapted from

4 The, split up the fractio ito + 5 > series, diverges. + 5 # Note that > +, so. Thus, by comparig this series with the diverget harmoic LIMIT COMPARISON TEST (Sectio.): Give three series, p, d, c : p if lim exists, ad c c is a coverget series, the p p if lim exists ad is ot equal to 0 or, ad d d the p diverges. coverges; is a diverget series, Similar to the Compariso Test, this test is also helpful whe cofroted with a series that appears u-testable. Most ofte, this test ca be applied to fractios with polyomials or with a combiatio of polyomials ad expoetials. Determie the covergece or divergece of the series Solutio: As icreases, the highest powers of the umerator ad the deomiator become the sigificat terms, i this case 2 ad. So which is a coverget p-series. Dividig the series by the reciprocal ) results i Now lim the series is a coverget series. 2 is similar to 2, 2 (i.e., multiply by. So, sice the limit exists, SERIES WITH BOTH POSITIVE AND NEGATIVE TERMS The ext sets of series tests are those that apply to series with both egative ad positive terms (kow as alteratig series). All the previous tests apply oly to those tests whose terms are all positive. A alteratig series is a series of the form #() + a a a 2 + a a + a 5 L, i other words, a series whose terms alterate betwee positive ad egative. Note that a is a sequece of positive umbers ad the () + determies the sig. Adapted from

5 ALTERNATING SERIES TEST (Sectio.5): For a give alteratig series, cosider a (the sequece of positive umbers without the () + ) if a + < a ad lim a 0, the the series coverges. This test will immediately give us the covergece of a alteratig series, just by lookig at its basic behavior. If each umber gets progressively smaller, ad the series approaches 0, the the series coverges. Determie whether the series ( ) # coverges or diverges. Solutio: For this series, the terms cotiue to decrease ad lim because these 2 criterio are met, the series coverges. 0 ad ABSOLUTE-CONVERGENCE TEST (Sectio.6): If a coverges, the so does a. Like the alteratig series test, this test checks oly for the covergece of a series ad says othig about the divergece. () 2 Does the series # coverge? ( + 2) 2 Solutio: Take the absolute value of the series, ad get 2. By ( + 2) 2 applyig the divergece test, this series diverges (the series cotiues to grow), so the absolute-covergece test is icoclusive. A later test might help solve this problem. A few defiitios will help clarify certai types of covergece. A coverget series A coverget series a is said to coverge absolutely if a coverges as well. a is said to coverge coditioally if a diverges. For example, the alteratig harmoic series coverges coditioally. RATIO TEST (Sectio.6): Give a series a, take the limit lim a + a if L <, the the series coverges; if L > or, the the series diverges. L This test says othig of the series if this limit equals oe. The ratio test ca be used i factorials ad expoetials Adapted from 5

6 Solutio: & Determie the covergece or divergece of the series $ # % Applyig the ratio test, we get lim + course, the geometric series test would also work here. Determie the covergece for the series,. '. < +. Thus the series coverges. (Of Solutio: This series has both a factorial ad a expoet. First, it is importat to + determie the + st term, which is a + ( +) Now, place it i the ratio test format, such that ROOT TEST (Sectio.6): a + a + ( +) ( +) +. So, lim 0 <, thus the series coverges. + For a give series: if lim # if lim # if lim # a a a L <, the the series coverges (i fact, it s absolutely coverget); > or lim # a, the the test is icoclusive. #, the the series diverges; This test will assist i determiig the covergece or divergece of moomials of degree, especially whe the term appears i the series. The oly terms that appear i such a series should be various degrees of aloe ad o other terms this test s effectiveess is limited to a small array of series. Determie the covergece of the series Solutio: Sice the series has both expoetials i the top ad bottom, the root test should be applied. So, 2 Thus, by the root test, this series diverges ad lim. UT Learig Ceter Jester A2A (86) 2/99 Uiversity of Texas at Austi