1. Do the following sequences converge or diverge? If convergent, give the limit. Explicitly show your reasoning. 2n + 1 n ( 1) n+1.

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1 Solutio: APPM 36 Review #3 Summer 4. Do the followig sequeces coverge or iverge? If coverget, give the limit. Eplicitly show your reasoig. a a = si b a = { } c a = e Solutio: a Note si a so, si coverges to by Squeeze Theorem. b Coverges to, sice lim + 6 / / el+6 a lim / L H el+6 c Note, sice +, a sice lim sequece coverges to by Squeeze Theorem. + Note that a e = a so the sequece e/+6 = e = e as, so lim a = e. = a thus the. For each of the followig series, etermie whether the series coverges absolutely, coverges coitioally, or iverges. Whe possible, state the sum of the series, Justify your aswers. + ta a + 5/ b c! + l = = = e l f l g + 4 h 4 Solutio: = a Diverges by Limit Compariso Test with b Absolutely Coverget by Ratio Test. = / = c Absolutely Coverget by Direct Compariso Test, sice < = ta + < = π/. a ote that sice the terms are all positive, a = a, thus covergece a absolute covergece are the same here Note that f = l is positive, cotiuous a ecreasig for 9 a we have 9 l }{{} = b u u l9 u b l9 b u u=l b l9 b b + l9 = l9 a so the itegral coverges a therefore by Itegral Test we have that the series is absolutely coverget. ote that sice the terms are all positive, a = a, thus the itegral test gives covergece, but absolute covergece are the same here e Coitioally coverget sice + l = l = l >

2 implies l iverges by Direct Compariso Test, a Series Test. f Diverget Telescopig Series sice = l = + = + l [ ] l l + = l lim l + = coverges by Alteratig g Geo Series absolutely sice terms are all positive, so a = a coverges to 8/3, here r = /4 <, a 7 4 = 7 7 = 4 /4 = 8/3 h Telescopig Series absolutely coverges to 3, ote that partial fractios, = 6 + = = 6 4 = = = 3 lim 5 5 a usig 3 + = 3 3. For the followig alteratig series, a! b = i. Show that the series is coverget. ii. How may terms must we keep i orer to approimate the sum to withi.? Note:. =. Solutio: a [i.] Use the Alteratig Series Test o the series, with b =! : Decreasig: For all >, we kow b + = + +! = + therefore the sequece {b } is ecreasig. Limit is : lim b! =. Therefore the series is coverget by the Alteratig Series Test.! = + b < b, [ii.] I orer to approimate the sum to withi, use the Alteratig Series Error Estimatio Theorem, which says error = S S b +, a here b + =. So to approimate + +! the sum to withi the esire error, we ee b + = + +! Now we just ee to pick large eough such that this is true. If you try =, the you ll get + +! = 6 = 6 >, but you ll fi = is ot large eough. Therefore = gives you the esire accuracy. b [i.] Alteratig Series Test, with b = : < + +! >. Decreasig: For all >, b + = + < = b, therefore {b } is ecreasig. Limit is : lim b =. Therefore the series is coverget by the Alteratig Series Test.

3 [ii.] Alteratig Series Error Estimatio Theorem: error b + = > = = < + + > Now, is certaily more tha 3 a less tha 4, so we shoul pick to make sure it is safely greater tha. So use 4 to get = 39. obviously, there s some wiggle room arou this 4. Give a 3 =! b + 3 c! 4 = i. For what values of oes the series coverge coitioally? coverge absolutely? ii. For what values of oes the series iverge? iii. Determie the iterval of covergece a the raius of covergece. Solutio: a i. No values of for which series coverges coitioally. Abs. cov. o,. ii. Not iverget for ay value of. iii. R.O.C = +, I.O.C =, b i. Coverges coitioally for =. Abs. cov o -,. ii., [, iii. R.O.C =, I.O.C = [-, c i. No values of for which series coverges coitioally. Abs. cov. at = 4. ii., 4 4, iii. R.O.C =, I.O.C is the sigle poit = 4. i. No values of for which series coverges coitioally. Abs. cov. o,. Note that, lim a + a = 4 < So, < 4 which implies < a so, the series coverges absolutely for < a iverges for > so R.O.C.=. At =, the series is At =, we have ii., ] [, = = iii. R.O.C =, I.O.C =,. +, which is iverget by Limit Compariso Test with. +, a this is iverget sice + iverges. 5. Evaluate the followig itegrals as power series. What is the raius of covergece? l 3 a b l Solutio: a = = = C = C +, so = =

4 l = C = C + C + + = C + C a we starte with a geo series a ever substitute for, so R = still. b 3 = 3 = 3 3 = = 3 = = = 3 4 = 3 + = 3 = + = C + +, so We substitute for i the geo series, so we ee < for covergece. Therefore < for covergece a R =. 6. Fi a power series represetatio for the followig fuctios. a f = Solutio: a + b l + = = C + + = = = b f = l + c f = = But pluggig i = implies f = = C + c = 5 5 = 5 = = C = C +, therefore C = a we get + + = = If you evaluate the itegral as a power series, how may terms are ecessary i orer to estimate the efiite itegral to withi the give accuracy? a, error. b l + 4, error. + 5 Solutio: a 5 = 5 = 5 = [ =, which is alteratig a coverget by the Alteratig Series Test, so the Alteratig Series Error Estimatio Theorem 5 + says error b + = = therefore we ee to pick at least 98 ]

5 b First, we ee a power series for l + 4, the we will itegrate it o [, ], assumig this iterval is withi the power series Iterval of Covergece. We kow l + 4 = , a the power series for + 4 = 4 is 4 = 4, so the series for l + 4 is = = 4 4+3, which we itegrate to fi l + 4 = = = C + we ca plug i = to l + 4 to fi C = = 4+4 +, where NOTE: We are oly allowe to move the itegral ito the summatio isie the Iterval of Covergece!! We kow this series for l + 4 is coverget for sure o 4 < < still, so the oly place we might be oig Magic Math is at the upper bou o the itegral, =. So we check if our power series for the itegra is coverget there: = gives: =, which is a coverget alteratig series by Alteratig + Series Test. Therefore we ca go ahea a itegrate: l+ 4 = = 4+4 = = Now this is a alteratig series with b =, which is both ecreasig a has limit Sice it satisfies the hypothesis of the Alteratig Series Error Estimatio Theorem, we kow error b + = = <. = > You will fi that = works here.