11.5 Alternating Series, Absolute and Conditional Convergence

Transcription

1 .5.5 Alteratig Series, Absolute ad Coditioal Covergece We have see that the harmoic series diverges. It may come as a surprise the to lear that ) )+ + = ) + coverges. To see this, let s be the th partial sum of ). The 2) s 2m = m 2m = ) ) m ) 2m Notice that each of the parethetical quatities is positive hece the sequece {s 2m } is icreasig. O the other had, 3) s 2m = 2 ) 3 4 ) ) 5 2m 2 2m 2m implies that s 2m. So that {s 2m } is a icreasig sequece that is bouded above. It follows by Theorem 6 from sectio 0. that this sequece has a limit, say lim m s 2m = L. Notice that Thus s 2m+ = s 2m + 2m+ lim s 2m+ = lim s 2m + lim m m m 2m+ = L+0 Combiig these results we coclude ) + = = lim ) + = = lim s = L Hece the alteratig harmoic series ) coverges.

2 .5 2 Remar. Later we will see two proofs that 4) ) + = l2 The previous example suggests the followig theorem. Theorem. The Alteratig Series Test Leibiz s Theorem) Let N be a positive iteger. The alteratig series 5) ) a = a 0 a +a 2 a 3 + =0 coverges provided that the followig three coditios are satisfied.. a > 0 for all N. 2. a a + for all N. 3. a 0 as. Proof. The proof is early idetical to the oe give above. Note: Coditio 2 is ot always easy to verify. Example. The alteratig series )+ / coverges by Leibiz s Theroem. To see this, observe that for all positive itegers ad 0 as. > 0 > +

3 .5 3 Defiitio. Types of Covergece A series a is said to coverge absolutely or is absolutely coverget) if the series a coverges. If the series coverges but ot absolutely, the the series is said to be coditioally coverget. Remar. The alteratig harmoic series is a example of a coditioally coverget series. O the other had, the alteratig p-series with p > is absolutely coverget why?). Oe must exercise care with coditioally coverget series as the followig example shows. Example 2. l2 = = = ) ) ) = = ) 2 = 2 l2 I light of the previous example, we agree always to Sum the terms of a coditioally coverget series i the give order.

4 .5 4 Theorem 2. The Absolute Covergece Test If a coverges the so does a. I other words, absolute covergece implies covergece. Proof. For x R we have x x x. Thus for each 6) a a a = 0 a + a 2 a Now the result follows by the Compariso Test. To see this, ote that if a coverges the so does 2 a. So because of 6) the Compariso Test implies that a + a ) must also coverge. Fially, a = a + a a ) = a + a ) a is the differece of two coverget series see Theorem 8 from 0.2 from the text). It follows by remars made at the ed of 0.2 that a coverges. The ext theorem explais why absolute covergece is preferred. Theorem 3. The Rearragemet Theorem for Absolutely Coverget Series If a coverges absolutely ad {b } is ay rearragemet of the sequece {a }, the b coverges absolutely ad a = b The proof is ot difficult.

5 .5 5 Example 3. Which of the followig series coverge? Which diverge? Justify your respose. If the series coverges decide whether the covergece is coditioal or absolute. a. ) + b. cosπ) c. 5) d. ) l l e. ) =2 l) 2

6 .5 6 Before we ca prove 4) we eed to prove a iequality that is importat i its ow right. Lemma 4. For x > 0 7) /+x) l+x) lx /x Observe that l+x) lx = ˆ x+ x dt/t So 7) follows from the setch below also, see the Max-Mi Iequality for Itegrals i chapter 5 of the text). y = /x x x+ x x+ Now let > be a iteger. The 7) implies 8) l+/) / l+/ ))

7 .5 7 Propositio 5. 9) ) + = l2 Proof. We have already show that the alteratig harmoic series is coditioally) coverget. Let be a positive iteger ad let S = = )+ /. We eed to show that S l2 as. Observe that S = = = ) = ) = It follows that 0) S = Exercise: Use iductio to prove 0).

8 .5 8 So by 8) ) l + ) l + ) Focusig o the left-had side of ) ad recogizig that la+lb = lab we see that The right-had side of ) reduces to l + ) = = l l = l + + = l ) + ) 2 + l + ) + = l +2 ) + = l2 Combiig these with ) yields l 2 ) + / l2 It ow follows by the cotiuity of the logarithm fuctio ad the Squeeze Law that lim S = l2. That is, ) + lim = l2 ad sice we must have = S + = S + + lim S + = lim S + lim + = l2+0 Together these imply 9).

9 .5 9 Example 4. Which of the followig series coverge? Which diverge? Justify your respose. If the series coverges decide whether the covergece is coditioal or absolute. a. ) +!) 2 )! b. ) + +/ c. =2 ) + l d. ) + +