Infinite Sequences. Ex.lim does not exist. n + 1 b) if the degree of the numerator is the same as the degree of the denominator, the. 1 n. Ex.

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1 Def: A ifiite sequece itegers. Exmple: UVW =,,,... Def: We sy the sequece RST Ifiite Sequeces l q is fuctio whose domi is the set of positive l q coverges to L if lim = L. Specil fcts bout determiig if sequece coverges: ) lim r = 0 if r < Ex. lim( / ) = 0 lim r r does ot exist if r > Ex. lim( 4 ) does ot exist ) For rtiol expressio (the quotiet of two polyomils p d q) ) if the degree of the umertor is greter th the degree of the deomitor, lim does ot exist. Ex.lim does ot exist. + b) if the degree of the umertor is the sme s the degree of the deomitor, the p lim equls the quotiet of the coefficiets of the highest degree terms. q Ex. lim = + c) if the degree of the umertor is less th the degree of the deomitor, the lim ) If lim is equl to 0. + = 0, the lim( ) = 0. If lim ot exist. + Ex. lim( ) = 0; lim( ) + + Ex. lim = = L where L 0, the lim( ) does does ot exist. + p q p q

2 Def: Let Ifiite Series l q be ifiite sequece. The expressio of the form = is ifiite series. = Ech ifiite series hs sequece S with it. l q clled the sequece of prtil sums ssocited Def: For the series = , the sequece of prtil sums S = ssocited with it is the sequece i which for ech, S is the sum of the first terms of the series; tht is, S = Def: The series = coverges d hs sum S if its sequece of = prtil sums coverges to S; tht is, if lim S Nth Term Test for Divergece: If lim diverge. (Note: Just becuse lim = S. 0, the the ifiite series = l q must = 0 does ot me tht the series coverges.) Usully we cot get S i form where we c directly fid whether or ot lim S exists, so we use certi tests for covergece of ifiite series to determie whether or ot series coverges. These tests my tell us tht the series coverges without tellig us the ctul sum. Tht is, they my prove tht lim S exists but ot tell us wht the limit is equl to. Two types of series i which we c directly cosider lim ) telescopig (collpsig) series Ex. [ ] = + S ) geometric series r = r = + r + r + r +... = ( + r + r + r +...) = = 0 A geometric series coverges d hs sum S = if d oly if r <. The r geometric series diverges if r. Be sure to write out the first few terms of the series, d if the first term is ot lredy, fctor it out to correctly idetify the vlues of d r, s we did i clss. Notice prticulrly whether (or whtever vrible is used) begis with 0,,, etc. Def: A p-series is series of the form, where p > 0. = p re:

3 Theorem: We c show by the Itegrl Test tht p-series coverges if p > d diverges if p. Ex.) = coverges / = = Ex. ) (clled the hrmoic series) diverges Ex. ) = = = diverges. / = Direct Compriso Test: Let ) If b for ll d = b = = d b = coverges, the ) If b for ll d diverges, the be positive term ifiite series. = b = lso must coverge. must lso diverge. Sice we kow exctly whe p-series d geometric series coverge d whe they diverge, we re most ofte comprig to these types of series, usully to p-series. We will most ofte use the Compriso Test whe is quotiet of terms of the form to costt power. Ex. Limit Compriso Test: Let = + + / + = d b = be positive term ifiite series. The if lim = k for some rel umber k > 0, the either both series coverge or both series b diverge. The Limit Compriso Test my be esier to use if you re ot sure wht iequlities you eed for the direct Compriso Test. To use the Limit Compriso Test, let be the th term of the series i questio, d geerlly we choose b by tkig the oe term from the umertor d the oe term from the deomitor tht gets lrgest s gets lrge. I this wy you should kow whether b coverges or diverges d lim should equl = b positive rel umber. Def: A ltertig series is series of the form = ( ) + = where > 0 for ll. 4 5

4 Altertig Series Test: The ltertig series ( ) + = coverges if: ) lim = 0 ) + < for ll = Ex. ( ) = coverges by the Altertig Series Test. Do ot try to pply the Altertig Series Test to positive term series. We lso kow the followig bout the sum of ltertig series: If S (the sum of the first terms) is used to pproximte the sum of coverget ltertig series, the error will be less th the bsolute vlue of the (+) st term of the series. Tht is, with coverget ltertig series S S <. + The Rtio Test: Let ) if lim + = <, the series coverges. be ifiite series. The + + ) if lim > or lim = the series diverges. ) if lim + =, the o coclusio c be reched by the Rtio Test. The Rtio Test is used if cotis fctor of the form " costt" to the power or fctoril expressios. + Ex. ( ) or 0 = =! The Rtio Test is lso used to fid the rdius of covergece of power series. Def: A power series i (x - c) is series of the form ( x c) = 0 + ( x c) + ( x c) +... = 0 For power series i x - c, exctly oe of the followig sttemets is true: ) The power series coverges oly for x = c. ) The power series coverges for ll rel umbers x. ) The power series coverges o itervl of rel umbers cetered t c; tht is, it coverges for x betwee c - r d c + r for some umber r clled the rdius of covergece.

5 Doig Problems o Covergece/Divergece Step. Ask yourself: Do I wt to determie if sequece coverges or if series coverges? For sequece, you just wt to kow if lim exists, d if so, wht is it? For series, go o to Step. Step : If you re tryig to determie if = coverges, check, if it is reltively esy to do so, lim. If this limit is ot 0, by the Nth Term Test for Divergece, the series = diverges. If this limit is 0, go o; you eed to use other test for covergece or divergece. (If lim esier wy to determie covergece or divergece.) does ot seem esy to fid, there my be Step : If the exct sum of the series is sked for t this poit i the course it is probbly either collpsig series or geometric series. I the cse of collpsig series, your write-up should show exctly wht S equls, the you should fid lim S, d you should use the defiitio of covergece of ifiite series. For fidig the sum of geometric series, idetify d r correctly d use the formul S = if r <. If ot oe of these kids of problems, go to: r Step 4. Decide if you hve ltertig series or positive term series. If you hve ltertig series, use the Altertig Series Test. If you hve positive term series, go to step 4. Step 5. If you hve positive term series, use the Direct Compriso Test, the Limit Compriso Test or the Rtio Test. If there is trig expressio or log expressio, etc. s fctor of, you my wt to mke compriso such s oe of the followig: e.g. si, > (sice l < ), sice si l si A compriso test is usully used if ech term i the umertor d deomitor is term of the form c where c is costt. For kid of series where you would use Compriso Test, you should be ble to tell whether you re pretty sure the series coverges or diverges before strtig your "proof" by tkig the quotiet of the "domit" terms i the umertor d the deomitor. If either the umertor or deomitor cotis fctoril expressios or term of the form c, you ofte wt to use the Rtio Test, but you my wt to mke compriso first. For exmple, < or <. It is esiest to use the Rtio Test! +! + whe there is ot sum i either the umertor or the deomitor.

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