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Unformatted text preview: (15) MA 166 EXAM 3 Spring 2002 Page 1/4 NAME QEADlIﬂBJEL STUDENT ID RECITATION INSTRUCTOR RECITATION TIME DIRECTIONS 1. Write your name, student ID number, recitation instructor’s name and recitation time
in the space provided above. Also write your name at the top of pages 2, 3, and 4. 2. The test has four (4) pages, including this one. . Write your answers in the boxes provided. 4. You must show sufﬁcient work to justify all answers. Correct answers with inconsistent
work may not be given credit. 5. Credit for each problem is given in parentheses in the left hand margin. 6. No books, notes or calculators may be used on this test. V DO 1. Circle the letter of the correct response. (You need not show work for this problem). 00
(a) Which of the following statements are always true for any series 21 an with
7),: positive terms? 00 00
(I) If lim an 2 0, then 2 an converges. Nat Lf‘U~£., Z J; Aiuzv es
n—)oo n=1 ht,
00
(II) If lim {Van = %, then Elan converges. True . Rodi: Zele
17,—)00 n:
n 5 0° . °°
(III) If lim “T = —, then 2 a... dlverges. 12...... com)...” Wu, .L
n—)oo — 6 n=1 1,:1 "(71
ﬂ wlm'dn oiz‘vnges m3. um. limit mmFarx‘liggn test  an+1 _ 0° .
(IV) If 11m — 1, then “21 an diverges. Not” tm 18‘ ' Rana tea}: {hwmmsgvt n—mo an J—
EN; anex»%€a A. (II) and (IV) only B. (I), (II) and (III) only C. (I) and (III) only E23
® (II) and (111) only E. all
(b) Which of the following series converge? (I) wnvevcécs. Co mParigoaum test) n ampules with 2’1 (I) Z W (H) 22 W n1! n'i/z CﬂHV£\"2e%, Ln'vm't Carrlloqn55o*rr Test
M ) (I) " with J5“
(111)1—3,1 +3i—31—+3i—... Wm m >
\/§ x/g x/ZI \/5 convex—ram. Altccvies Test A. (I) and (II) only B. (I) and (III) only C. (II) and (III) only D. (III) only [:7] ® all MA 166 Exam 3 Name Spring 2002 Page 2 /4 (20) 2. Determine whether each series is convergent or divergent. You must show all necessary
work and write your conclusion in the small box. For Problems W0») amt} 2(5) QM“ WYSI it"
I} Wrowﬁ‘ "9 O P?) [AV Frckl‘am
1 " check work We ‘CN'Z‘ mCD .L
YI to" V.
0“ div. ()i__—1__
a “:1 e/n(n+1)oz+2) Show all necessary work here: {D diverges (P58ri:C)s/ ’32)) VIII W hm,
lln (n+4) (\M 2) h—pm ' 1
3“] n Z
1 TrO By the Rim i 1 co mpouﬁ son test, the series is (LL Day. 00.: n1; By the \(‘qx‘m‘r’ test MA 166 Exam 3 Spring 2002 Name _________._______ Page 3/4 00
_1 n——1
(10) 3. Consider the convergent alternating series n.
n=1
(a) Write out the ﬁrst six terms of the series.
1 ’ 23+":1i 4% ‘5! 63 Row
L a, ,L l. . /
1  i. + «a m ~—« Jr“ —’' ' ——‘
2 G 2.4“ 120 7‘20 (b) Find the smallest number of terms that we need to add in order to estimate the sum of the series with error < 0.01. (3)
_L *< 0. or @
12° 4 m
A, > o, o I “’ 24 oo
“1 71—1
(10) 4. Determine whether the series 2 (——2— is absolutely convergent, conditionally n=1 nﬁ convergent, or divergent. You must justify your answer. . v «mloiem . (9) 5. Find the sum of each series if it is convergent, or write divergent in the box. No partial
credit. on <> i 1 h i “L‘”:‘L” ———’“* O
a e :— ’_ 2’3— (in 71" 3
":1 34:1 9’1) e '92 e 1 <b>§2:<%>" : Ba— <33 MA 166 Exam 3 Spring 2002 Name Page 4/4
69 Jud) c' (LVN. Von s" n (A
Pnigsinék \qere (xx«A QP’aQAYS lei6" 0‘“ 00
(16) 6. Find the interval of convergence of the power series Zn3(x — 5)". Don’t forget to 71:1
test for convergence at the end points of the interval. You must show all work. Palm Jich 'x«5 Ham “‘5‘”
ﬁ' two la. mi
—: W (MTVBl : \x—sl @ warms?
wowm n u'U‘u) Kn wmhg
h Swim umeﬁcgm PM lstkl Ox“ 4<2< <6 C2)
00
n 3 . C13 . .
WW X 24’ i Z n 0h¢€ﬁ%€f7 teal” @f;ﬁ;ﬂ$§mﬁa
r55 3 1 ® ﬂ )
WM x :6 'o E W dlmyg 17?: (1) $0) .
, ' a a. (9 g [C0
.‘ In/iUUaL 0/ CAYIVCAKBQ/M . y<‘4) he ow ln: 4 .
(10) 7. Evaluate the indeﬁnite integral / 1 +11: 4 (1:1: as apower series and determine its radius
of convergence R. 0° 0° 00 '
i 1x :2 E: x" ) ma, .1. : (x*)":Z<—D“x4”
1’ ":0 1+)t4 hTO mo /
_ 00 4
1 n 4h (l Or‘ \X‘4l
Sm4x:g(ZO(—HX )d~.x lXi
n: .
w y, 441+.
_ _ X
__ C +. n}; D m 2 [x<L (10) 8. Find the Taylor series for f 2 371—1 centered at a = 1. "" (U V” 1.
i”) z 2. gm“) (x4) : H1) + Lei1&4) + :59! 6(~1)+~~ n! K ﬂ 1. ...
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 Mathematical Series, recitation instructor, Recitation Time, following series converge, limit mmFarx‘liggn test

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