Istructor: Math 0560, Worksheet Alteratig Series Jauary, 3000 For realistic exam practice solve these problems without lookig at your book ad without usig a calculator. Multiple choice questios should take about 4 miutes to complete. Partial credit questios should take about 8 miutes to complete. PLEASE MARK YOUR ANSWERS WITH AN X, ot a circle!. (a) (b) (c) (d) (e) 2. (a) (b) (c) (d) (e)... 3. (a) (b) (c) (d) (e) 4. (a) (b) (c) (d) (e)... 5. (a) (b) (c) (d) (e) Please do NOT write i this box. Multiple Choice 6. 7. 8. 9. Total
Multiple Choice Name: Istructor:.(6 pts) For what values of p is the followig series coverget? ( ) l. p =2 (a) p > 0 (b) for all p (c) for ay p such that p 0 (d) p > (e) p < 0 Solutio Sice this is a alteratig series, We oly eed to apply the alteratig series test. If l p > 0 the b + < b, ad lim = 0 if p > 0 ad = if p < 0, so the aswer is c. p 2.(6 pts) The series ( ) 4 2 is a alteratig series which satisfies the coditios of the alteratig series test. Use the Alteratig Series Estimatio Theorem to determie the smallest k o the list below so that the k-th partial sum is withi 00 of the actual sum. Solutio. Via the Alteratig Series Estimatio Theorem we kow the error of the kth partial sum is bouded by the (k + )st term b k+ so we wat 4 (k + ) 2 00 which meas k 9, hece the best possible choice is 20. (a) 25 (b) 5 (c) 0 (d) 20 (e) 50 2
Istructor: 3.(6 pts) The series 3 ( ) is a alteratig series which satisfies the coditios of the alteratig series test. Fid the smallest umber k o the list below so that the k-th partial sum is withi of the actual sum., 000 (a) 5 (b) 25 (c) 0 (d) 20 (e) 50 4.(6 pts) A series coverges coditioally if the series coverges but the sum of the absolute values of the terms does ot coverge. Which series below coditioally coverges? Solutio. Recall that a series is coditioally coverget if it is coverget but ot absolutely coverget. Note immediately that a) ad c) are diverget as their terms ted ot to zero as goes to ifiity. Now, b), d), ad e) are coverget by the alteratig series test. Further, cosiderig the correspodig series give by takig the absolute value term wise we see that d) ad e) are absolutely coverget, while b) is ot. (a) ( ) (b) ( ) 2 (c) ( ) e (d) ( ) 3 (e) ( ) 3
Istructor: 5.(6 pts) A series coverges absolutely if the sum of the absolute values of its terms coverges. Whe a series coverges absolutely it coverges. The series ( ) + =2 (a) (b) (c) (d) (e) coverges absolutely. diverges because the terms alterate. ( ) + diverges eve though lim = 0. does ot coverge absolutely but does coverge coditioally. ( ) + diverges because lim 0. 4
Istructor: Partial Credit You must show your work o the partial credit problems to receive credit! 6. (0 pts.) Show that the series ( ) 2 + 3 9 is coverget usig the Alteratig Series Test. The series ( ) 2 + 2 + is alteratig ad lim = 0 sice it is a limit of a ratioal fuctio for which the degree of the umerator is less tha the degree of the deom- 3 9 3 9 d x2 + iator. Fially, sice x 3 9 = 2x(x3 9) (x 2 + )(3x 2 ) = x4 3x 2 8x < 0 for dx (x 3 9) 2 (x 3 9) 2 x > 0, the sequece 2 + is decreasig. Hece the Alteratig Series Test shows ( ) 2 + 3 9 3 9 is coverget ad therefore coditioally coverget. 5
Istructor: 7. (0 pts.) (a) Cosider the series ( ) l. Fill i the followig blaks =3 ad be sure to show your work. I each case idicate which test you are usig ad show how it is applied. Is the series coverget? (YES or NO) Y ES We apply the Alteratig Series Test: l lim = lim / = 0 (usig L Hospital s Rule). We use the derivative of f(x) = l x x to show that a + < a for 3. f (x) = l x x 2 < 0 for > 3 > e. Therefore a + < a for 3 ad the series coverges by the Alteratig Series Test. Is the series absolutely coverget? (YES or NO) NO a = l. This diverges by direct compariso with the p-series l, sice > for all 3 (b) Suppose you wat to fid a approximatio to the sum of the above series usig k a partial sum S k = ( ) l. Use the Alteratig Series Estimatio Theorem to determie the umber of terms of the series you eed to add such that the error S S k < l(99.5) 99.5. We eed k with a k+ < l(99.5) l(k + ) or 99.5 k + is decreasig for x > 3, we kow that l(00) works. 00 < l(99.5) l x. Sice the fuctio f(x) = 99.5 x < l(99.5). Hece k + = 00 or k = 99 99.5 6
Istructor: (l )2 8. (0 pts.) Cosider the series ( ). Fill i the followig blaks =3 ad be sure to show your work. I each case idicate which test you are usig ad show how it is applied. Is the series absolutely coverget? (YES or NO) Aswer is No. Note that (l )2 ( ) = (l ) 2. For 3 we kow that =3 =3 (l )2 l which gives. Recall that the series diverges. By the Compariso Test, diverges as well. =3 (l ) 2 =3 Is the series coverget? (YES or NO) This series is a Alteratig series. There is a possibility that we ca use Alteratig Series Test. (l )2 Let b =. The b > 0. Now, (l ) 2 (l x) 2 lim = lim [Where x is a real umber.] x x = lim 2 l x [Usig L Hospital] x x = 2 lim = 0 x x [Usig L Hospital] If we ca show b is decreasig the we ca use the Alteratig Series Test. We will use (l x)2 Calculus I to see if b is decreasig. Let f(x) =. The f 2 l x (l x)2 (x) = = x x 2 l x(2 l x). For large x we have 2 l x < 0. This shows that f (x) < 0 for large x x 2 ad hece f(x) is decreasig for large x. (l )2 Now by Alteratig Series Test, ( ) coverges. =3 7
Istructor: 9. (0 pts.) Test the followig series for covergece. Specify the exact test beig used, ad check that all the required hypothesis are satisfied. (a) ( ) + Sol.(b) We ote first ad clearly + > 0. Now let f(x) = x x +. lim + = 0 f 2 x (x) = (x x + ) 2 Whe x > 4, f (x) < 0. So f(x) is decreasig whe x > 4. So by alteratig series test the series i questio is coverget. 8
Istructor: ANSWERS Math 0560, Worksheet Alteratig Series Jauary, 3000 For realistic exam practice solve these problems without lookig at your book ad without usig a calculator. Multiple choice questios should take about 4 miutes to complete. Partial credit questios should take about 8 miutes to complete. PLEASE MARK YOUR ANSWERS WITH AN X, ot a circle!. ( ) (b) (c) (d) (e) 2. (a) (b) (c) ( ) (e)... 3. (a) (b) ( ) (d) (e) 4. ( ) (b) (c) (d) (e)... 5. (a) (b) (c) ( ) (e) Please do NOT write i this box. Multiple Choice 6. 7. 8. 9. Total