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Unformatted text preview: Prof. Girardi Math 142 Spring 2010 03.18.10 Exaﬁ
MARK BOX PROBLEM POINTS
1 a — j 30 _.__¢'  " "2==:.:%. rixq‘: (1) To receive credit you must: Xxx”
(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 lin‘e/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. If you have a question, raise your hand. When you ﬁnish: turn your exam over, put your pencil down, and raise your hand.
(5) This exam covers (from Calculus by Stewart, 611 ed, ET):
11.2~11.8 . Problem Inspiration: Mostly homework and old exam problems. See the solution key for details. 00
1' Fillinthe blanks / boxes. All series 2 are understood to be n21
Hint: I do NOT want to see the words absolute nor conditional on this page! 1a. nthterm test for an arbitrary series 2 an.
If limnnoo an 7E O or lirnn_>00 an does not exist, then 2 an Cl We"? 9? 1b. Geometric Series where —oo < r < 00. The series 2 r” o converges if and only if lrl < i o diverges if and only if {TI 2 l 1c. p—series where O < p < 00. The series 2 n—lz;
7 l
. . , < I
o diverges if and only if p — o converges if and only if p 1d. Integral Test for a positive—termed series 2 an where an 2 0.
Let f: [1, co) —> R be so that ,an=f(f\ )foreachnEN
o f is a funcmon
. f is a (33; its??? function . j .7 4
Then an converges if and only if i. TU )“X A” converges. 1e. Comparison Test for a positivetermed series 2 an where an 2 0. 0 HO 3 an 3 bn for alln E Nanden Carve‘98 ‘4'" ,then'Zan Cm
7» oIfOSbnganforallnENanden diwafﬁev ,thenZan ‘7' a/CWfi 63 1f. Limit Comparison Test for a positivetermed series 2 an where an 2 0.
Let bn > 0 and lirn,H00 = L. \ 1“ {wk 1 If Q___ < L < G3 . 7 then 2 an converges if and only if . 1g. Ratio and Root Tests for a positive—termed series 2 an where an 2 0. Let p = limnnoo “2:1 or p = limnnoo (an)%.
o If p C l then 2 an converges.
0‘ If p 7 i then 2 an diverges.
o If p : then the test is inconclusive. 1h. Alternating Series Test for an alternating series Z(—1)”an where an, > 0 for each n E N.
If
a an > an+1 for each n E N
o lirnnnoo an = 0 then Z(—1)”an is "V a" ‘3: ‘3 3‘ 1i. By deﬁnition, for an arbitrary series 2 an, (ﬁll in the blanks with converges or diverges).
0 2 an is absolutely convergent if and only if 2 anl COMET$8?
0 2 an is conditionally convergent if and only if 2 an CQW‘PWQT and 2 ion] '3 3 0 2 an is divergent if and only if 2 an lj. 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 2701:” an is: absolutely convergent,
conditional convergent, or divergent. Does Z [an converge? if No Does
=> if NO 2a,, is
limnnmlan[=0? => ‘ Since {an} 2 0, use a positive term test: {an WW6” integral test, CT, LCT, ratio/root test.
if YES ll  if YES ll Ea is madden Cygr‘v'é‘lfjggz‘l Is 2% an alternat
TL _—____________— ing series? if YES ll Does 2 an satisfy the conditions of the Al
ternating Series Test? if YES ll 2. Circle T if the statement is TRUE. Circle F if the statement if FALSE. T ® If lirnnnoo an = 0, then 2 an converges
(T) F If 2 an converges, then limnnoo (1,, = 0.
@ F If 2 an converges and 2 b7, converge, then Em” + bn) converges.
T {f F If 2(an + bn) converges, then 2 an converges and 2 bn converge.
K.
1/ _ N 7‘ _ TNll
1 _ n _ F IfSN—Zr,thenSN——1——_—r—.
‘E n21
‘4} 4W“ "3
.1," ~. _
€ h} 2 { ﬂ; \, , ,9
, g It} 2; d K l ’
WMaWMwwmw wmmwm’““m‘ 3. Check the Correct box and then indicate your reasoning below. Speciﬁcally specify What tes:t(s)
you are using. A correctly checked box Without appropriate explanation will receive no points, W convergent °° (1)”
Z n E conditionally convergent I M"‘ \. pie!”
w" 1" \  ~— l \‘1 i ?\
PVT“, rm
A‘s”?
’i—x
{A , L 7 “‘ :Uh+\
r h 'er, h
V (“Kali
R M (f C31
m 2:20 sow W Z n r‘ .2”
F4303 4. 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. .\
absolutely convergent 00
1
Z —————————— D conditionally convergent
71:1 \/ n (n+1) (n+2)
\ D divergent
on 1 W i
Z s 2‘ If; ,3 :qh
b'—\ \ erixﬂrlﬁ it ' “oi:
LCT \
bf“ " K “’5\ «‘3 ‘
_ ’5 p 5/ 1
i { r\.
“9“” a? “kw J ' l oz ‘r V0 l
i m ,f , (’59 a” \ \>“u< V ‘3’ K iv)" bf)...” CkLI‘lk \r4333 3   w, (j r
 "w.  W“, W “0‘ J a .1 "
‘2 : xv“? M E E 3 2 lﬂr ‘ 2:; 53‘)" ,, " Fab.“ f ‘ or V V a 'if” ' ‘ .3,» i \,, , K
T WWW t 912*? (of
'7‘ , ’5 _ q, ,3, ‘ J” r { g m » l , .
C_,~’;r‘~§ 5 ’ (C kg ‘3!“ “" ‘ € 9395 \fr {r “Blend 5. Let
n! (2n — 1)! an: 5a. Find an expression for “2” that does NOT have a fractorial sign (that is a ! Sign) in it. TL 5b. Check the correct box and then indicate your reasoning below. Speciﬁcally specify What tesc(s)
you are using. A correctly checked box Without appropriate explanation will receive no points. absolutely convergent 6. Consider the formal power series n
n=1
Hint: (5m + 10)“ = [5 (m + 2)]n = 5n (cc + 2)” = 5” (a: — (—2) )n  j,
The center is $0 = ‘5 ‘ and the radius of convergence is R = 5 __ . As we did in class, make a number line indicating Where the power series is: absolutely convergent,
conditionally convergent, and divergent. Indicate your reasoning and speciﬁcally specify What test(s) you are using. Don’t forget to check the endpoints, if there are any.
(Ar hulkgratin A _ I I. ; ‘ . _ f .1»
(it 1.1 Wim a» 1 x w. a”: 32m “ v : W, E!
i
t '
C ,4 \ r” l v Y ’A (I) ‘3,
.s
i A. . . f1:
: z E ' f “ ‘
1‘ u 1 I .1
r E ‘ {I a); 4 h L f h ‘3 A‘ v r l s
.. : % ‘», {A “ l l ' : ‘ “ l
f “WW 3. a: r
r.
. .,
5 _ . g“
i Mi ....... u I [6 wwwmt
K C30.» 4—H“ a}
m n ‘2‘ l
i kn": w 3+ .4; A r». w EAL/ii, {La
; . x _ A ~ I;
; W t,” g f ?'\ ‘wl " : (“*2
g: i at i
)2. M!) V: \l
u‘ " ‘l V}
i .... , r »* ”" ”” f r
, [l “K 'x Wax; x ’ i 1” f V3
E, a, J a V I“ 7. Consider the formal power series i W
Tl
TF2 (1n n)
Hint 1: (1%?? = so would you rather use the root test or the ratio test? Hint 2: ln(ar) rln(a) but (in (a))r # rln(a) + The center is 1130 = O and the radius of convergence is R = Cm
As we did in class, make a number line indicating where the power series is: absolutely convergent,
conditionally convergent, and divergent. Indicate your reasoning and specifically specify what
test(s) you are using. Don’t forget to check the endpoints, if there are any. (j, \30 A} i F L 'Q lv\ (7:) ”r yrs {,«c 3 5»: g" A 2f _, ,. n/ v / Eng—r {a} a.» V lE r/
X h l I“, H“ VI \ii n‘ N 1 ) O : I)
J? 1’1“ _..__ h F ﬂ’l if” b Egg? “E F b Fill—in the 6 blanks. Consider the power series 00 Z (—1)” anxn n=1 where all of the an’s are positive. Let‘s say that you know that if
if
if 0<zc<17
$217 17<$ Then this power series has: center at 3:0 = Also, what can you say about the following interval? Fill in the blanks below with: o is absolutely convergent o is conditionally convergent is divergent then :X—l)”L anxn' converges
then Z(—l)” anx" conditionally converges
then 2&1)” aux” diverges . l7 and radius of convergence R = inconclusive (not enough information given to decide in general). : 2 g , {if if  17 < m < 0 then 2H)” It a
if x < —17 then Z(_1)n anmn , a r
1f ‘xZO then ZlDn arm” 0mm” H“ <4“ Vela?"
m V .5,"
,'~’€L»n€_ he ,
if a: = ——17 then Z(—1)” 0mm” (i i i J
, s h ‘ I
Z w; I : ﬂ W \ 5‘ P
2: { i; any”) 9. Geometric Series. Let, for N _>_ 102, N
2”
SN 2 Z 37n+1 '
712102 9a. Do some algebra to write 5 N as 25:102 c r" for an appropriate constant c and ratio r. 9b. Using the method from class (rather than some formula), ﬁnd an expression for 5N in closed form
(i.e. Without a summation 2 Sign nor some dots 9c. Does 22:102 3% converge or diverge? If it converges, ﬁnd its sum. Justify your answer. ...
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 Calculus

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