Math 113 Exam 4 Practice

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1 Math Exam 4 Practice Exam 4 will cover.-.. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for you to work o. The third sectio give the aswers of the questios i sectio. Review Series I this sectio we leared about coverget ad diverget series. A series coverges if the sequece of partial sums coverge. There are some particular types of series that we leared about: Geometric Series ar a. We leared that the geometric series coverges to if r < ad diverges otherwise. We saw several applicatios where we could write a problem i terms of a geometric r series. Harmoic Series. We saw by examiatio of the partial sums s that this diverges. ( ) + Alteratig Harmoic Series We kow by a later test that the alteratig Harmoic series coverges. We kow by the Maclauri series of l( + x) that it coverges to l(). Telescopig Series This is a series where the partial sum collapses to the sum of a few terms. We ca the take the limit of the partial sum to see what the series coverges to. Note that i. the oly series whose sums we could calculate were geometric ad telescopig. Tests for Covergece We leared about the followig tests for covergece: Divergece Test If a 0 the a diverges. This is a excellet test to start with because the limit is ofte easy to calculate. Keep i mid, however, if the limit is 0, the the Divergece test tells you othig. You must try some other test. p series If you recogize a series as a p series, the you ca use the fact that a p series coverges whe (ad oly whe) p >. Geometric series We discussed this i the last subsectio. Compariso Test To use the compariso test, we eed to have a large group of test series available. We also eed to kow if these test series coverge or ot. The most commo test series for the compariso test are the p series ad the geometric series. If the series acts like a p series, or acts like a geometric series, the you may wish to use the compariso test. Remember, if 0 a b ad b coverges, the a coverges. a diverges, the b diverges. Limit Compariso Test This test works well for the type of problems that also work with the compariso test, but is somewhat easier. You still eed the test series, but you do t eed to work to make the terms of the series greater tha or less tha some kow series. You oly eed to check the limit p a lim. b If it is fiite ad positive, the both series coverge or both diverge. Sice you already kow about oe of them, you the kow about the other.

2 Itegral Test If we are tryig to determie whether a coverges, ad there is a fuctio f(x) with f() = a, the the sum coverges iff a f(x) dx. (We assume that both the series {a } ad f(x) are positive.) So the itegral test is hady if the associated fuctio ca be itegrated without too much difficulty. Alteratig Series Test To use the alteratig series test, you eed to verify three thigs: The series is alteratig. (This ca usually be doe by ispectio). The terms of the series coverge to 0. (Hopefully you did this whe you applied the Divergece test.) Fially, the terms of the absolute values are decreasig. The secod statemet does ot ecessarily imply the third. If this is true, the the alteratig series test tells us the series coverges. Ratio Test If lim a + a = L, the the series is absolutely coverget if L < ad diverget if L >. If L =, the test fails. This test works really well whe a factorial is preset i a. Root Test If lim a = L, the the series is absolutely coverget if L < ad diverget if L >. If L =, the test fails. This test works really well whe there are powers of i a. Remember, the Itegral test ad the compariso tests oly work whe the series has o-egative terms. If you have a series where the terms are both positive ad egative, the you must be able to say whether the series coverges absolutely, coverges coditioally, or diverges. It is oe of these. These are mutually exclusive coditios. Power Series A power series is a fuctio defied by series: f(x) = a (x c). c is called the ceter. Oe of the questios we ask about a power series is: Where does it coverge. We eed to fid the radius of covergece ad the iterval of covergece. To fid the radius of covergece, use the ratio test. lim a + (x c) + a (x c) = L x c The radius of covergece is R = /L. To fid the iterval of covergece, you eed to test the series where the ratio test fails: at c R ad c + R. Note that the above formula may covice you that you oly eed to fid L, which is the ratio of the successive terms. However, it is better to use the ratio test outright. For example, suppose we wish to fid the radius of covergece of (x ). The ratio test tells us that which must be less tha to coverge. Hece, but the radius of covergece is ot /. It is /. lim + (x ) + (x ) = lim x, x <,

3 Fidig sums of series Fidig a power series that represets a specific fuctio is the ext topic. The first oe we leared was the geometric series: x = x. We the foud the sum of several series by differetiatig, itegratig, multiplyig by x, etc. The Taylor series of a fuctio is f () (c) (x c)! ad ca also be used to fid the power series of a fuctio. Notice that the iterval of covergece of these series is still very importat. We eed to kow whe we ca trust them. I additio to the geometric series above, the followig Maclauri series (with iterval of covergece) are importat: ta x = l( + x) = ( ) + x+, (, ] ( ) + x, (, ] e x ( ) r =! x, (, ) ( + x) r = x, see book. If you eed to costruct a Maclauri series of a fuctio ad some of the above fuctios are icluded, it is almost always easier to maipulate the Maclauri series istead of costructig the series by scratch. Approximatig sums of series si x = cos x = ( ) ( + )! x+, (, ) ( ) ()! x, (, ) I additio to fidig whether sums of series coverge or ot, we also were able to fid approximatios to the error. There were basic approximatios to the error give by the Itegral test, Alteratig Series test, ad the Taylor Series.. If f() = a ad f is a cotiuous, positive ad decreasig fuctio for x k, the k+ f(x) dx =k+ k f(x) dx. If ( ) a is a alteratig series where {a } is a positive decreasig sequece with a limit of 0, the k ( ) a ( ) a = ( ) a < a k+. =k+. If T (x) is the th Taylor polyomial of f(x) cetered at c, ad R (x) is the remaider, the o the iterval where f (+) (x) < M. R (x) M x c + ( + )! We use this iformatio, whe applicable, to fid maximum errors whe approximatig a fuctio by a Taylor polyomial as well.

4 Questios Try to study the review otes ad memorize ay relevat equatios before tryig to work these equatios. If you caot solve a problem without the book or otes, you will ot be able to solve that problem o the exam. Determie whether each sequece i to 4 is coverget. State what it coverges to, if applicable. Is the sequece icreasig or decreasig? Is the sequece bouded?. a = a = cos(π/). a = si + 4. a = +. Determie whether the series is coverget or diverget. If it is coverget, fid its sum = = k k k 5 Determie whether each series i questio 8 to coverges or diverges. If the series alterates i sig, state whether it coverges absolutely, coditioally or diverges. State ay covergece/divergece tests you use. e ( l k= k= k= ) ta k + k k (k + ) k l k l() = cos + ( ) l ( )! cos(π) + ( ) ( + ) ( ) (l ) ( ) 5 +. Show that 5 is a upper boud o the error of 4 if the sum is approximated by the first + 7 two terms. 4. Suppose the power series a (x+) has a radius = of covergece R = 5. List all possible itervals of covergece. 5. Fid the radius ad iterval of covergece of (x ) 6. Fid the radius ad iterval of covergece of ( 4) (x ) + 7. Fid the radius ad iterval of covergece of x 8. Fid the radius ad iterval of covergece of x 9. Fid the radius ad iterval of covergece of! x Fid a power series represetatio i powers of x for the fuctio f(x) = + x with iterval of covergece.

5 . Fid a power series represetatio i powers of (x ) for the fuctio f(x) = +x ad give the iterval of covergece.. Fid a power series represetatio i powers of (x ) for l( + x).. What is the power series represetatio of x ( x)? 4. Fid the Maclauri series for f(x) = l( x) from the defiitio of a Maclauri series. Fid the radius of covergece. 5. Fid a Taylor series for f(x) = cos(πx) cetered at x =. Prove that the series you fid represets cos(πx) for all x. 6. Use multiplicatio to fid the first 4 terms of the Maclauri series for f(x) = e x cosh(x). 7. Use divisio to fid the first terms of the Maclauri series for g(x) = x cos x. 8. Use the power series of +x to estimate. correct to the earest Justify that the error is less tha usig the Alteratig Series Estimatio Theory or Taylor s Iequality. 9. Fid the sum: (a) (b) = ( ) ( ) + (+)! (c) ( ) x (+)! (d) 4! + 8! + 6 4! + 5! Fid the Taylor polyomial T (x) for the fuctio f(x) = arcsi x, at a = Approximate f by a Taylor polyomial with degree at the umber a.ad use Taylor s Iequality to estimate the accuracy of the approximatio f(x) T (x) whe x lies i the give iterval. (a) f(x) = x, a = 8, =, 7 x 9; (b) f(x) = x si x, a = 0, = 4, x. 4. Fid the Taylor polyomial T (x) for the fuctio f(x) = cos x at the umber a = π/. Ad use it to estimate cos 80 0 correct to five decimal places. 4. A car is movig with speed 0m/s ad acceleratio m/s at a give istat. Usig a secod-degree Taylor polyomial, estimate how far the car moves i the ext secod. Would it be reasoable to use this polyomial to estimate distace traveled durig the ext miute? 44. Show that T ad f have the same derivatives at a up to order. Aswers. coverges to 0, decreasig, bouded. diverges, ot icreasig or decreasig, bouded.. coverges to 0, ot icreasig or decreasig, bouded. 4. diverges, icreasig, bouded below. 5. diverges by the Divergece Test 6. Coverges to / (Telescopig sum) 7. Coverges to 5/4 (geometric series) 8. Use the itegral test x e x dx = 5 e Therefore it coverges by the itegral test 9. Use the itegral test ( ) l x dx = Therefore it coverges by the itegral test. x 0. Use the itegral test ta x π dx = + x Therefore it coverges by the itegral test. x (x + ) dx = 9 Therefore it coverges by the itegral test.. We have to be careful here sice the fuctio is ot defied at k =. By a chage of variables, k = + we see that k l k = ( + ) l( + ) ad k= we ca the use the itegral test. The book otes that we ca also simply chage the limits of itegratio, though it does ot state this as a theorem. dx = ( ) (x + ) l(x + ) Therefore it diverges by the itegral test < + 6 = ( ) ad ( ) coverges (geometric r = ). Thus + 6 coverges by Com- pariso Test.

6 4. lim Hece Compariso Test. l 5. < = () Thus l() ad = ad diverges. diverges by Limit diverges (p-series). diverges by Compariso Test. cos 6. + < + < ad coverges (p-series ). Thus coverges by Compariso Test. cos + 7. Coverges by the Alteratig Series test. By the Itegral Test, it does ot coverge absolutely. So it coverges coditioally. 8. Diverges by the Test for Divergece. 9. Coverges by the Alteratig Series test. By the Limit Compariso test (with b = ), it does ot coverge absolutely. So it coverges coditioally. 0. Coverges absolutely by the Limit Compariso test (with b = ).. Coverges by the Alteratig Series test (Use L Hôpital s rule). By the Itegral Test, it does ot coverge absolutely. So it coverges coditioally.. Coverges absolutely by the Limit Compariso test (with b = / ).. Sice 4 +7 < it is sufficiet to show that 4 5 is a boud o the sum. The R 4 x dx = (-6,4), (-6,4], [-6,4), [-6,4] 5. R =, I = (, ) 6. R = /4, I = ( /4, + /4] = (7/4, 9/4] 7. R =, I = (, ) 8. R = /, I = [ /, /] 9. R = 0, I = {0} x = ( x = ( x ) = ( ) x. + x = = + (x ) = ( ) (x ). (x ). Itegrate the above to get. Sice Thus, ( ) ( + ) (x )+ = ( ) d = dx x ( x), ( ( x) = d ) x = dx x ( x) = x ( ) + (x ) () x. x = x l( x) = l() + ( )!! x, R =. 5. cos(πx) = ( ) + π ()! (x ) 6. e x cosh(x) = + x + 5 x + 6 x g(x) = x cos x = x /6 x 4 / x = x + 4! x 4 7! x +... so. = withi Because the series is alteratig the error for this sum is less tha tha 8 the ext term, which is , which is less tha Fid the sum: (a) ( ) ( = ) + (+) ta ( = π 6 (b)! = e (c) ( ) x (+)! = si(x) x (d) 4! + 8! + 6 4! + 5! +... = e 40. T 5 (x) = x + 6 x 4. (a) T (x) = + (x 8) 88 (x 8), Error is withi ; (b)t 4 (x) = x 6 x4, Error is withi 0.04; 4. T (x) = (x π ) + 6 (x π ), cos 80 0 = m, No 44. Prove by mathematical iductio or directly cosider the k th derivative of the polyomial T.

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