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Unformatted text preview: Math 122 Test 3 November 21, 2006 1
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Name \<\ ll"; ‘1' 6
Style Points
Total g
Directions:
1. No books, notes or losing to the Steelers in the last minute. You may CI! use a calculator to do routine arithmetic computations. You may not
use your calculator to store notes or formulas. You may not share a
calculator with anyone. . You should show your work, and explain how you arrived at your an— swers. A correct answer with no work shown (except on problems
which are completely trivial) will receive no credit. If you are not sure
whether you have written enough, please ask. . You may not make more than one attempt at a problem. If you make several attempts, you must indicate which one you want counted, or
you will be penalized. Numerical experiments do not count as justiﬁcation. For example,
computing the ﬁrst few terms of a series is not enough to show the
terms decrease. Computing the ﬁrst few partial sums of a series is not
enough to show the series converges or diverges. Plugging in numbers
is not enough to justify the computation of a limit. You may, of course
use numerical experiments to help guide your work, but they do not
count as justiﬁcation for answers. On this test, explanations count. If I can’t follow what you are doing,
you will not get much credit. . You may leave as soon as you are ﬁnished, but once you leave the exam, you may not make any changes to your exam. The exam has 6 questions and a total of 100 points. Happy Thanksgiving 1. (15 points) (21) Find the limit of the following sequence: { (2:1)”? 11:1
 x , “x ‘ V W N «i {N N W N
w gin ma I 3 x ,_ N, WWWWWWWW .3 7 'w JV“fo was; il ) “I “’1‘?”va ., EX) l
H “:3” (if) N “‘5 (7.; “v _ m
V i l i [K M VA,
A t in M x r in l we WW ,,,,,,,,,,,, x A Oz «N .%M::£:L.:L3{ril :: 51:) s mi N «H 7W”’fi{“‘”“‘"“ _l
N «em N we WTTTW ”””” s2 is. W was M
N m: x (b) Determine if the following series converges or diverges7 and if it
converges, ﬁnd the sum: 00 J5 n 9 n. .r . ":4. memw,
'. w t, “x “v” Z‘j'TQ“ Cg»  rm;
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fawn“: T”? i” W M Ma if:
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3, 2. (20 points) For each of the following series, determine if it converges or
diverges. For each test you use7 you must name the test, perform the
test, and state the conclusion you reached from that test. 00
1 n . ., 21‘ —~——~—,—~ DW M9 ewe,
( ) mum/2)
«.017 ‘ ‘y ‘  { a) ‘ {3% If“ EM 42;}
m “ﬁrearm.. "a" ( W3 AAAAA M j X cwm w r, , f M
Te??? {i eat wow >2 ‘ « Qt (b) i (7103 51C} N; ezgggéﬁg M, f\ ’ . . S ‘ ‘ W M M A "rm: ‘ “d
R ééK/VY‘} i Q j}: {‘3 M59 (J: N :K k X :33ng C 6‘“) “1” J}
"T5351 we, cm (W W)? (k m g 00 2 s‘n2n w r. K n“??? 3?:
((1) Z Cami: wAii mm.) r“ k a K #3:) q \ [I N " h""’vrww
_ r. .\ 2.1%» “is \N N g “fryW ems w a < t“ Fm ‘
C53 m g) m: t .2 CM “MWMW “* ~' « t
"322’ g” M N? "a: t ,3 3. (20 points) Determine if the following series converge absolutely: con—
verge conditionally, or diverge. For each test you use, you must name
the test, perform the test, and state the conclusion you reached from
that test. 00 2 F”
n + 4n + o 3; gig) wax»;
d) ~__1 n ‘ ‘u M N m M1
()g( )3n2+3n+2
\ more r , v _
liwﬁf k “gag—yaw“ ‘ﬂjmmjj_:miwm “3‘— :E: Q
(s3 l ‘3' "a ‘ ~ ‘ ~ 2. _ ‘ 5;», a ““ Kris“ 4: all
00 (_2)n “ E t W; M, 1 CE. R Q 6:“
(b) 2 n, (draw \reléi.>t Mi? l M; J a
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W Marl; K J?
a o \
WM/ “nix ” a , \a
it???“ N Asa Ce» U“ “H 1"? 654::
“I ‘ m “f”‘””"“"‘““ A ‘
g" m ML, all >1 ~: gel, rwa l 33” V
a 7 J a ~~~~~~ ~~ 53¢
lwferﬁmm W a» “it liar“ >5 v g ‘5 WWW
"' WW“ 3 V W WWWWW : "K k MW «mmmw w» w A M
lat 3. " lwr @733“ {J “1/ \ r“7“”*M§”3 M g4 “tar: Ma ““ H») g Winnl,
00 I 8271 (A {D ,r
d ~1 7“ “5(ng \1 “2:;
< > ( > gamma: wwwwwwwww 7 p
f" \ E g) HEAR) f") W W
WW WQMWX ” 1‘1“ “ W N Q/Wn w 4. (15 points) Consider the power series: (3 — f6)”
n + 1 n22 Where is the power series centered? '25 W C (b) Find the radius of convergence. N‘t} (rang) w») A \gmbd <1 (I '2, QM W M "’r .3}
N “5% CC} (C) Find the interval of convergence. [Hintz Check the endpoints.] f, '~ ' " x ML” bx v. L.“ CT W 6:“ xix“: \ N :7 El
5 N CK! * é LN C‘QN‘J CQND a N. me i N ‘3: 31 SE 5. (15 points) For
f (93) = 0080112) (a) W'rite out the ﬁrst 4 non—zero terms of the Madaurin series for ﬁx)
4 l4 5;
ELK L“ (GA
Li 9 hi
Cog) (' :1“ i “— 3:” “t mg...“ W W " “ "
2% I4 2_ (a) (1)) Find P4 (an), the fourth degree Madaurin polynomial for f w  m. A»
we \ a 4
,2 :1:
cos :1: — 1 +
(C) Compute 11m —(——>—8——~—~_2:
m——>() 33 You must ShOW your work in order to receive credit. 3 I X ,
a a; e
/“ gwawi ~ m/e/{f
ELM A “ESQ; (o L / i 6. (10 points) Indicate whether the following statements are true or false
by circling the appropriate letter. A statement which is sometimes true
and sometimes false should be marked false. 00
If 0 < an < b” and Z an converges7 T e3 then 2 bn diverges.
n:1 If the nth partial sum of the series 2 an is @ F 00
Sn : yL—Z—Z, then 2 an 2 1 7L:1 00 CO
If the series 2 an diverges and the series Z bu diverges, 71:1 'nzl 00
then the series Em” + bn) diverges. 11:1 T® WE 00 If lim an 2 0 then E an converges.
’(‘lf—FOO 1
n: 00
If Z on converges then an 2 0. ® F
n:1 ...
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This note was uploaded on 02/17/2010 for the course MATH 122 taught by Professor Butler during the Spring '07 term at Case Western.
 Spring '07
 Butler

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