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Unformatted text preview: _________________.___——
Prof. Girardi Math 142 Fall 2011 10.18.11 Exam 2 NAME: K C¥ PIN: INSTRUCTIONS: (1) To receive credit you must:
(a) work in a logical fashion, show all your work, indicate your reasoning;
no credit will be given for an answer that just appears;
such explanations help with partial credit
(b) if a line/ box is provided, then:
— show you work BELOW the line/box
— put your answer on/in the line/ box
(c) if no such line/box is provided, then box your answer
(2) The MARK Box indicates the problems along with their points.
Check that your copy of the exam has all of the problems. (3) You may not use a calculator, books, personal notes. (4) During this exam, do not leave your seat unless you have permission. If you have a question,
raise your hand. When you ﬁnish: turn your exam over, put your pencil down, and raise
your hand. (5) If you do not make at least 17.5 out of the 35 points on Problem 1, then your score for the
entire exam will be whatever you made on Problem 1. (6) This exam covers (from Calculus by Stewart, 6th ed., ET):
11.2—11.7 . Honor Code Statement
I understand that it is the responsibility of every member of the Carolina community to uphold and maintain
the University of South Carolina’s Honor Code. As a Carolinian, I certify that I have neither given nor received unauthorized aid on this exam. I understand that if it is determined that I used any unauthorized assistance or otherwise violated the
University’s Honor Code then I will receive a failing grade for this course and be referred to the academic
Dean and the Ofﬁce of Academic Integrity for additional disciplinary actions. Furthermore, I have not only read but will also follow the above Instructions. Signature : C
Source VPTHLQ 51a“ 02 4&— ‘ él 7—
1' Fillinthe blanks/boxes. All series 2 are understood to be n=1 1a. 1b. lc. 1d. 1e. 1f. 1h. nthterm test for an arbitrary series 2 an. If limnqm an 75 0 or limnnoo an does not exist, then 2a,; Cl ll/dra e 5 . Geometric Series where —oo < r < 00. The series 2 r" «A.
2:1. 0 converges if and only if lrl
o diverges if and only if lrl pseries where 0 < p < 00. The series % o converges if and only if p >
4 2
1L
o diverges if and only if p L Integral Test for a. positivetermed series 2 an where a." 2 0.
Let f: [1,oo) —) IR be so that 'an=f(_n_ )foreachnEN "'JV/noranreasmj
o fisa pOSl‘t‘iVC gﬂﬂ function (LS also 0&2
o f isa reajin function 0 fisa. COYH'MIJQLLS function. co
Then 2a,. converges if and only if S 1 £02 dz converges. Comparison Test for a positive—termed series 2 an where an 2 0. (Fill in the blanks with a" and/or 1;...) o If 0 s an S bn for all n E N and 2 bn converge, then 2 an converge.
o If 0 S b" S a." for all n 6 N and 2 bp diverge, then 2 an diverge. Limit Comparison Test for a positive—termed series 2a,. Where an 2 0.
Let bn > 0 and limn_,oo ﬁg = L. b
If 0 < L < 00 then 20,; converges if and only if Z I“ comm rg¢ 5 ——5 . Ratio and Root Tests for arbitrarytermed series 2 an with —00 < an < 00. gm P=1imn~oo Let p = lirIin_.oo a" or
o If p < \ then 2a,, converges absolutely.
o If p > l then 2a” diverges.
o If p —' I then the test is inconclusive. Alternating Series Test for an alternating series E(—l)"‘aﬂ where an > 0 for each n E N. If
> an+1 for each n E N
o limnnoo an = D then Z(—1)"an COMBO} {.5 can 1i. By deﬁnition, for an arbitrary series 2 an, (ﬁll in the blanks with converges or diverges).
« 2 an is absolutely convergent if and only if 2 Ian] C 0 7W ‘5 I" C S
0 2a,, is conditionally convergent if and only if 2 a,, CO “Verfaﬁand E a.,, Adages 0 2a,. is divergent if and only if 2a,, diverg 85
1j. Fill in the 3 blank lines (with absolutely convergent, conditional convergent, or divergent) on the following FLOW CHART for class used to determine if a series 22:17 (111 is: absolutely convergent,
conditional convergent, or divergent. Does Z a,, converge?
Since Ianl Z 0, use a positive term test:
integral test, CT, LCT, ratio/root test. Is 2a,. an alternat
ing series? ifYES 1} Does 2 an satisfy the conditions of the Al
ternating Series Test? ZanisMMMgAﬁ 1k. Circle T if the statement is TRUE. Circle F if the statement if FALSE. l T I F If 2 an converges, then lin1,,_.,.,‘D a1, = 0.
T CF) If lining“, an = 0, then 2 an converges
@ F If 2 an converges and 2 bn converge, then 2(an + b,,) converges. If 2(0.n + bn) converges, then 2 an converges and 2 bn converge. N _ N
T ® If SN = Zr", then SN = r T , for when r aé 1 sincewedon't llkato dividebyurc 1 — 'r
1L=l Source. a, Warm w}? "(Ldvxls'olch hullalmle T‘Ol’lﬂm. ’/ 2. Check the correct box and then indicate your reasoning below. Specifically specify what test(s)
you are using. A correctly checked box without appropriate explanation will receive no points. E absolutely convergent °° (1)"
2 D conditionally convergent '5, L4)
n l A/ ? 2’: >l
Zl nL Z n C/o’vw. Since. WV; 2: (:22 AL» alas.ch SW SLWFIhMMoMA‘p/(Lm 3. Check the correct box and then indicate your reasoning below. Speciﬁcally specify what test(s)
you are using. A correctly checked box without appropriate explanation will receive no points. [I absolutely convergent [Z] divergent You may use, without showing, that for n 2 1 4n7—n6+2l big, RN; 5,, PaginVc—n‘umol sane; _ 4n7—n6+21
“n=m>°'
Hint. What would be easier: CT or LOT?
'4
71mm Liv—aw "5% ill, , LL 1,;
__________..... ,
Lew1L Ilng—n’rn ”’ “A? “
.L.
30 by» Jaw LCT “’1 En: n
1r (1 C6 vtlln .300 Ll
a“ Wn—n+ll.ﬂii;ﬂ”.br
L H g_{\+r’+ I IerLnH'l' 0V vL . Comma/ES 6n, it 3. m caﬂwllaﬂo‘ms ow +114, I'D/IOVViHj W w 4 m,
j.. < “Min 4—2!
Hm ‘ 'lng—n+l1 n’’
W
4n?’nb+1l Lf
q E —— <7;> We 2 Abba. q
’ <
“7,\ Mn 45) ‘+(Hn<3'nkl?) < H“ (4h7—hbi’1‘3 <=> an—Llwrw 4 L’L‘ng "'Y‘? ‘“ 33‘“
q.
4.)} 0 < 'lln man»?
but 7900 : "Hvx +235ex'é8
Vu)= ’11m“~+23E
__ 5
\c'ogzo es» 'Xa « Z” if; 3 Li
$30 _,(__,_‘C_,,<£_.
,. x0
#(xo) ’3 Ilw M) ’ '°°
7v?!)
#0) = '59
HQ) :: —;z)o(,
/ 4 4
Jew—{:13 292), ’2 m,
H(2)‘31+—+ ’ agar '” 4b. Check the correct box and then indicate your reasoning below. Speciﬁcally specify what test(s)
you are using. A correctly checked box without appropriate explanation will receive no points. ,3 absolutely convergent a: (—1)n (Mn—l 1)! [:l conditionally convergent
D divergent
(nHll _ (n+0 ! (ml Cougar Mames: ” MT”? = M. “P
(1 (2M2. ll (ZnH).
,i—P—"w‘d'ﬂ/f l(n+l LZW' . l _ . ,__,_——————
amt : (“MN .(ln—lﬂ i : Ely—’2 ' (“val (2n)(2n+)
"If m “HT” ’ M. (ml)! “‘ ‘ 0
(MI) n+\ ﬂ %%5:0=€<\
:: : / 7' 2.;
M Lin“ Jan 1 Ll“ n L 5. Check the correct box and then indicate your reasoning below. Speciﬁcally specify what test(s)
you are using. A correctly checked box without appropriate explanation will receive no points. {:I absolutely convergent El conditionally convergent llWL 11L...— :: ‘75 H.900 3n1+g“,\
livl AV
vow. 237?:— 3'
Solvﬁnﬂkwwultvll” 3’ 3"l
in m“ 3n HZn l
“$00
n “Lkl Hmwﬁg 'PFOIOILM g “57’ 44; I3 6. Check the correct box and then indicate your reasoning below. Speciﬁcally specify what test(s)
you are using. A correctly checked box without appropriate explanation will receive no points. D absolutely convergent 00 I
Z (—1)" H” E conditionally convergent
n=1 V n D divergent J Lu: , 2—~ln
19. Lctf(:r)=—,_.Thcnf(z)= I<thenlnz>2orz>e2,soﬂ m I [,2 ﬂ is decreasing fem > 8.
.  , . Inn . 1/ I
BylHospxtalsRulc 11m — = n — lim 2 ‘ m In”
“M” n” x _ __ = 0' m _ "_ 
‘6; a: 1/ (2 v/E) mm «a so 653168,§l( 1) J; converges bytnc Alternating Series Test Limit Comparison Test Let b“ > 0 and link—0° if = L. his ‘ (LCT) [f 0 < L < co, then [ E a." conv. ==~ :bn conv. ] (you DO need to memonze I one:
M 1H; 2 I If L = 0, then [ :bn conv. = S on conv. ] (you do not have to memorize this one:
/ ,4 If L = 00. then [ ‘7‘ b... divg. = E a." divg. ] (you do not have to memorize tins one) . I.
Let in = L um :19“ 3ng Ha. p—iews) Pwﬁ l, 8 unﬂiﬂAcﬂ, EOMI/OJ’ be Sum 31 it? 31mg 90/0 7. Geometric Series. (On this page, you should do basic algebra but you do NOT have to do any
fancy arithmetic (eg, just making up numbers, you can leave @9171 as just that.) Let, for N 2 51, 3n+1 N
ans/=2:257I . n=51 7a. Do some algebra to write 3N as 2§=51 c r" for an appropriate constant c and ratio r. 7b. Using the method from class (rather than some formula), ﬁnd an expression for 3N in closed form (i.e. Without a summation 2 sign nor some dots .). ...
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 Fall '11
 KUSTIN
 Calculus

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