Review of Sections

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1 Review of Sectios.-.6 Mrch 24, 204 Abstrct This is the set of otes tht reviews the mi ides from Chpter coverig sequeces d series. The specific sectios tht we covered re s follows:.: Sequces..2: Series, geometric series d telescopig series..3: The itegrl test d estimtig sums..4: The direct compriso test..5: The ltertig series test..6: Absolute covergece d the rtio d root tests..: This sectio covers sequeces of umbers. To begi we recll the defiitio of sequece, which is ifiite list of umbers {, 2, 3,...} = { } =. We re iterested i the limit of the sequece, which if it exists we deote by We lso gve the followig forml defiitio. lim = L. Defiitio. The limit lim = L if for every ɛ > 0 there exists N such tht L < ɛ wheever N. As exmple we cosidered = +. Note tht we c evlute the limit directly s folllows lim = lim + = lim + / =, implyig tht L =. Now we cosider the forml defiitio L = + = ( + ) + = + < ɛ.

2 To show tht the limit of is L =, we eed to show tht we c fid N such tht + < ɛ whe N. This will hold if we fid N such tht + N + < ɛ. Solvig for N we hve N + > ɛ or N > ɛ. Now, if we re give ɛ, the we choose N such tht N > ɛ, which will imply + N+ < ɛ. We lso hve the followig theorem tht llows us to use the limit of fuctio f(x) to evlute the limit of sequece. Theorem. Assume tht lim x f(x) = L d tht f() =. The lim = L. Limit lws for sequeces: we hve the followig lgebric idetities for limits of sequeces. Let { } d {b } be two give sequeces of umbers d c costt. The, lim c( b ) = c lim c lim b, (0.) x x x where = +,,,. I dditio, we hve the Squeeze theorem for sequeces. Theorem 2. The squeeze theorem for sequeces. Assume tht b c for some 0. If lim = lim c = L the lim b = L. The theorem lso implies tht if lim = 0, the lim = 0. Filly, we hve the Mootoic Sequece Theorem. Theorem 3. Mootoic Sequece Theorem. Assume tht is mootoic sequece such tht + > (or + < ) for some 0. If is bouded (m < < M) the lim coverges..2: Itroducito to geerl series, geometric series d telescopig series. This sectio cocers series, or ifiite sums of the form i. i= To evlute the series or determie if it coverges or diverges oe ofte works with the -th prtil sum S = i. This leds to the series {S } of -th prtil sums, whose limit gives the sum of the series: lim S = lim i= i = i= i. If the series sums to fiite umber we deote its sum by S d sy tht it is coverget series with S = i. We lso defie the geometric series d stted coditios o its covergece. i= 2 i=

3 Theorem 4. The geometric series. The geometric series + r + r 2 + r = r i, > 0 i= is coverget for r < d diverges otherwise. If it coverges, the its sum is give by r i = i= The proof is obtied by cosiderig the differece r. S rs = + r + r r (r + r r + r ) = r. Thus, implyig tht wheever r <. S ( r) = ( r ) or S = ( r ), r lim S = r, Additiolly, we hve the followig theorem tht c be used to test for divergece. Theorem 5. The divergece test. If i= i is coverget, the lim i i = 0. Thus, if lim i i 0 or the limit does ot exist, the i= i diverges. Lws for series: we hve the followig lgebric idetities. sequeces of umbers d c costt. The, c( i ± b i ) = c i ± c b i. i= i= i= Let { } d {b } be two give Filly, we looked t telescopig series i which we used prtil frctios to evlute S d thereby the sum of the series, for exmple, Tkig the -th prtil sum we hve i= i(i + ) = i i +. S = (/ /2) + (/2 /3) + (/3 /4) (/( ) /) = /. Thus, lim S = lim / =..3: The itegrl test d estimtig sums. The itegrl test is quite geerl techique for determiig the covergece of series d lso i estimtig their sum with prtil sums. The followig theorem summrizes the result. i= 3

4 Theorem 6. The itegrl test. Assume tht f(x) is cotiuous, positive d decresig fuctio o some itervl [, ), > d let f(i) = i. The, i= i coverges if d oly if f(x)dx coverges. Applyig this result to p-series of the form i= i, we sw tht the p-series coverges for p > p d diverges otherwise. Note tht we c ot evlute the series usig the itegrl test, we c oly determie its covergece or divergece. To estimte the sum of the series, we use the followig result tht chrcterizes the error i pproximtig the series with the th prtil sum S. Theorem 7. Deote the error i pproximtig series with the th prtil sum by r = S S = The, uder the ssumptios tht f(x) is cotiuous, positive d decresig fuctio for x, f(i) = i d tht i= i is coverget, we hve + f(x)dx r f(x)dx..4: The direct compriso d limit compriso tests. Theorem 8. The direct compriso test. Let { i } d {b i } be two give sequeces such tht 0 i b i for ll i. The, i i i b i d. If i i diverges, the i b i diverges. 2. If i b i coverges, the i i coverges. Two results tht re of use here re. The p-series i= i p coverges for p > d diverges otherwise. 2. The geometric series i= ri, > 0, coverges for d r < d diverges otherwise. I certi cses it is more coveiet to use the limit compriso test, which follows from the bove direct compriso test. Theorem 9. The limit compriso test. Let { i } d {b i } be two give sequeces such tht 0 i, b i. Assume tht i lim = c > 0, (c < ). i b i The,. i i d i b i both coverge, or 2. i i d i b i both diverge. 4

5 .5: The ltertio series test. Theorem 0. The ltertig series test. Cosider the ltertig series If. b i+ b i for ll i d 2. lim i b i = 0, ( ) i b i, b i > 0. i= the the ltertig series is coverget. Estimtig sums of ltertig series simplifies from the geerl cse, s the ext theorem shows. Theorem. Estimtig sums of ltertig series. Assume tht the ltertig series ( ) i b i i= stisfies items () d (2) i the ltertig series test so tht it is coverget. The, r = S S = b + (b +2 b +3 ) (b +4 b +5 )... b +, sice we ssumed tht b i+ b i. Hece, the error i usig S i pproximtig S = i= ( )i b i is bouded by the ( + )st term, b +..6: Absolute covergece d the rtio d root tests. Defiitio 2. A series i= i is bsolutely coverget if i= i coverges. If i= i is coverget but i= i is diverget, the the series is sid to be coditiolly coverget. Note tht bsolute covergece implies coditiol covergece sice i i i i. The rtio d root tests re the two mi techiques for determiig bsolute covergece. We summrize the results i the ext two theorems. Theorem 2. The rtio test. Let lim i i+ i. L <, the i i is bsolutely coverget, 2. L >, the the series is diverget, d 3. L =, the the rtio test is icoclusive = L <, L > 0. If Theorem 3. The root test. Let lim i ( i ) i. L <, the i i is bsolutely coverget, = L <, L > 0. If 2. L >, the the series is diverget, d 3. L =, the the rtio test is icoclusive Note the root test d rtio tests will both be icoclusive whe such series rises. Thus, if you pply the rtio (root) test d fid tht L =, the you will get the sme results for the root (rtio) test. 5