The Comparison Tests. Examples. math 131 infinite series, part iii: comparison tests 18

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1 math 3 ifiite series, part iii: compariso tests 8 The Compariso Tests The idea behid the compariso tests is pretty simple. Suppose we have a series such as which we kow coverges by the p-series test. Now compare this to the series. The terms i this ew series are smaller tha the correspodig terms the first series + + sice 0 < + + <. So the sum of + + should be smaller tha the sum of. Sice the latter series coverges, so does the former. There are some techical details that eed to be checked for istace, the terms of the series eed to be o-egative. But this idea ca be made ito a proof which we will omit here. The result is THEOREM 4.6 (The Direct Compariso Test). Assume that 0 apple a apple b for all (or at least all k). ote: As stated, this test requires 0 apple a apple b for all. But this coditio. If b coverges so does a. may be relaxed so that 0 apple a apple b for all > k.. If a diverges so does b. The way to thik about this theorem is if the bigger series coverges, so does the smaller oe. If the smaller oe diverges, the so does the bigger oe. Pre-law ad pre-med studets (i additio to math studets) should delight i usig the the direct compariso test because oe eeds to see a patter ad the costruct a little argumet. I will be lookig for these argumets whe I grade your work. Examples To use the compariso test effectively, you eed to kow lots of series that diverge or coverge to which you ca compare a ukow series. Such series are ofte provided by such tests as the p-series test, the geometric series test, or eve the itegral series test. Let s see how this works. EXAMPLE 4.. Does + coverge? SOLUTION. scrap work: Notice this ot a series to which the itegral test easily applies, or is it a p-series or a geometric series. However, it looks a lot like the geometric series which coverges. Whe we compare a series to a covergig series, we wat the ukow series to be smaller tha the kow series so that we ca use the the first part of the the direct compariso test to show that the ew series also coverges. I this case otice that 0 apple + < for all. OK, let s give a careful argumet. argumet: Sice 0 apple + apple for all, ad sice coverges by the geomet- ric series test (Theorem 3.) because r = <, the direct compariso test (Theorem 4.6). coverges by the +

2 Notice the argumet is ot log but it has two importat aspects. First we idetified a related series that we kew about for compariso. Secod we verified the appropriate hypothesis relatig the terms of the ukow series to the oe we kew about. EXAMPLE 4.6. Does l coverge? SOLUTION. scrap work: We could use the itegral test. However, this looks a lot like the harmoic series which diverges. Whe we compare to a divergig series, we are expectig the ukow series to diverge so we wat the terms of the ukow series to be larger tha the terms of the kow series. But is it true that l >? OK, let s give a careful argumet. argumet: Notice that I particular, if l 3, the l () l () e.. Sice diverges (p-series test with p = ), l the diverges by the direct compariso test (Theorem 4.6). Notice how we justified the steps i the argumet, eve justifyig why we kow diverges. EXAMPLE 4.7. Does + 6 coverge? SOLUTION. scrap work: This looks a lot like the geometric series which coverges. argumet: Sice 0 apple +6 apple for all, ad sice geometric series test (Theorem 3.) because r = <, the the direct compariso test (Theorem 4.6). EXAMPLE 4.8. Does! coverge? = 9 coverges by the coverges by + 6 SOLUTION. scrap work: Notice that the terms of this series get small very quickly. So we should suspect that it coverges. This ot a series to which the itegral test easily applies, or is it a p-series. It takes a bit of algebra to see what to compare it to. argumet: Notice! = ( ) ( ) =.So0<! apple. Now we could apply the itegral test to the series to see that it coverges, ad the use the direct compariso test to see that! coverges. But we ca avoid the itegral test by usig a bit more algebra. Notice that () () (). So this meas! if. So! <, whe. However coverges (p-series, p = > ) so by direct compariso (Theorem 4.6) the series coverges.!

3 math 3 ifiite series, part iii: compariso tests 0 YOU TRY IT 4.6. Each of the followig statemets is a attempt to show that a give series is coverget or diverget usig the Compariso Test. Classify each statemet, correct if the argumet is valid, or icorrect if ay part of the argumet is flawed. (Note: Eve if the coclusio is true but the argumet that led to it was wrog, classify it as icorrect. ) (a) For all > 3, 0 apple apple l(), ad the series =3 Test, the series =3 l() diverges. (b) For all >, 0 apple p < + p + Test, the series diverges. ad the series diverges so by the Compariso diverges, so by the Compariso (c) For all >, 0 apple <, ad the series 3 3 coverges, so by the Compariso = Test, the series = 3 3 coverges. (d) For all, 0 apple cos () 3 coverges, so by the Compari- 3 so Test, the series cos () < 3, ad the series (e) For all, 0 apple < +, ad the series 3 Test, the series The Limit Compariso Test 3 coverges. Aswers to you try it 4.6 : (a) Icorrect: 6apple. (b) Correct. l() (c) Icorrect. 0 6apple 3 3 (d) Correct. (e) + 3 coverges. coverges, so by the Compariso While the direct compariso test is very useful, there aother compariso test that focuses oly o the tails of the series that we wat to compare. This makes it more widely applicable ad simpler to use. We do t eed to verify that a apple b for all (or most). However, it will require our skills i evaluatig its at ifiity! THEOREM 4.7 (The Limit Compariso Test). Assume that a > 0 ad b > 0 for all (or at least all k) ad that a = L.! b () If 0 < L < (i.e., L is a positive, fiite umber), the either the series b both coverge or both diverge. () If L = 0 ad b coverges, the a coverges. (3) If L = ad b diverges, the a diverges. a ad a The idea of the theorem is sice = L, the evetually a Lb. So if oe! b of the series coverges (diverges) so does the other sice the two are essetially scalar multiples of each other. Icorrect. If a series is larger tha a covergig series, the compariso test does ot apply. EXAMPLE 4.9. Does coverge? SOLUTION. scrap work: Let s apply the it compariso test. Notice that the terms are always positive sice the polyomial has o roots. I ay evet, the

4 terms are evetually positive sice this a upward-opeig parabola. If we focus o highest powers, the the series looks a like the p-series which coverges. argumet: Sice the terms 3 +6 ad are positive, we ca apply Theorem 4.7. a 3 +6! b! =! =! = 3 > 0. Sice coverges by the p-series test (p = > ), the coverges by the it compariso test (Theorem 4.7). ote: Whe the series ivolve fractios, the first step i the it process ca be doe more efficietly. Istead of dividig oe fractio by the other, we ca multiply oe fractio by the reciprocal of the other. For istace, earlier i this example we could have writte a! b! ad the carried out the rest of the calculatio. is called the ge- c + d EXAMPLE 4.0 (The Geeral Harmoic Series). The series eral harmoic series. If c > 0 does this series coverge? SOLUTION. scrap work: Let s apply the it compariso test by makig the obvious compariso to the harmoic series which we kow diverges. argumet: Sice the terms c+d are positive oce c + d > 0, i other words whe > d c, ad sice ad is always positive, we ca apply Theorem 4.7. a! b! c + d =! c + d = c > 0, sice c > 0. Sice diverges by the p-series test (p = ), the the geeral har- moic series EXAMPLE 4.. Does the series diverges by the it compariso test (Theorem 4.7). c + d p coverge? SOLUTION. scrap work: This time if we focus o highest powers, p 3 +4 ad are always positive, we ca apply Theo- / roughly equal to / 3 =, which diverges / argumet: Sice the terms rem 4.7. p is a = /! b! / = 3! =! + 4 = > 0. 3 Sice / coverges by the p-series test (p = p > ), the coverges by the it compariso test (Theorem 4.7). EXAMPLE 4.. Does the series p 4 + coverge? SOLUTION. scrap work: The obvious compariso is to the p-series diverges. which /

5 math 3 ifiite series, part iii: compariso tests are always positive, we ca apply Theo- argumet: Sice the terms p ad 4+ rem 4.7. / a p / HPwrs = / p = /! b! 4 +! 4! / = > 0. Sice / diverges by the p-series test (p = < ), the the series p 4 + diverges by the it compariso test (Theorem 4.7). EXAMPLE 4.3. Does the series coverge? SOLUTION. scrap work: Focusig o highest powers, 4 = argumet: Sice the terms 6 which we saw is diverget i Example is roughly + 3 ad are always positive, we ca apply Theorem 4.7. (Note the use of the 4 +3 reciprocal.) a 6 =! b! =! =! Sice diverges from Example 4., the the series it compariso test (Theorem 4.7). EXAMPLE 4.4. Does the series = 6 > ( ) ( l(e )+) cos(p) 3 coverge? diverges by the SOLUTION. scrap work: First simplify the th term: cos(p) =( ) right? Ad l(e )=. So ( ) ( l(e )+) = +. Use a it compariso test with cos(p) 3 3. argumet: Sice the terms ( ) ( l(e )+) = + ad are always positive, we cos(p) 3 3 ca apply Theorem 4.7. a +! b! 3 = 3 + HPwrs! 3 = 3! 3 = > 0. Sice coverges by the p-series test, the the series ( ) ( l(e )+) cos(p) 3 coverges by the it compariso test (Theorem 4.7). EXAMPLE 4.. Does the series cos coverge? SOLUTION. scrap work: The terms cos are always positive. Use a it com- pariso test with. argumet: Sice the terms cos ad are always positive, we ca apply Theorem 4.7. cos cos si x = l Ho x = x 3 =!! x! x! x x 3 a! b si x = 0. Sice coverges by the p-series test, the the series cos coverges by the secod part of the it compariso test (Theorem 4.7).