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Unformatted text preview: 813 D Math 122 Test 3 — V1 ER April 7, 2009 Name—Ki—Y Directions: 1. No books, notes or 6 inches of snow in April. You may use a calculator
to do routine arithmetic computations. You may not use your calcu
lator 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. . You may leave as soon as you are ﬁnished, but once you leave the exam, you may not make any changes to your exam. 1. (10 points) (a) Compute the limit of the following sequence: W9};
QQW N6\N(‘k‘73 2% giNFﬂ) N¢300 ~\—~ Nﬁoo N ~‘ (135(4)
QM 752 “‘ = 1.
N500 «jg (b) Determine if the following series converges or diverges, and if it
converges, ﬁnd the sum: 71.20
W G3!
\ \)
ng‘ﬁx ‘“ é?“ CON
N20 3 “:0
i ,_ a~§ 2.1,
X _ _____._ — a Z;
\‘J. i’é ' Raw 0 EST CO W1 P
TESW/ 2. (20 points) For each of the following series, determine if it converges or
diverges. For each test you use, you must name the test, perform the
test, and state the conclusion you reached from that test. 002”ng TL g“ m s2 Q; “'m oo
:2 °° 1
(c) ;%COS(E)
 \ COMV
l , _I_ . l 5 NC f”?
Vcﬂwl <t’1 ‘ “ “
“é MEG
sex
((1);;
n=1
52m .1. ~— 44w
5 ES
N‘Ew 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.
°° 1
(a) Z(—1)“n(hm)3 Com. MB _
71:2
1 0mm” C1100)
INrEeQaL. X—m = . 3, y _ /
TE? xwmq one. (as; \ .03
CK) ) ~ ~—
I (ﬁx: 3ng C:\0’‘"/CL— /1\
, , 2:: a0 ‘3
oo __1n
0» 2:;
”=2 \ >) i; burr—5 R91“
COWS?“ pm 2.17 N
l ll use)
(d) g—DHTSZB
‘ l’ 1 M
Reno .Q/Wm '5 (““3“) L M2} “ 2 <1
Tm xxx—3a) “fr ( H 4. (15 points) Match each of the following intervals of convergence With
the correct power series: a) [1,3] __C_L__ i (x _ 2)” Ex ) 5) mm L $3523)" E) a c)[—1,5) _Q___ i($“2)n E335) d) [1,3) L £9312)” D 355:} e>‘[1,51 5. (8 points) For f(:z:) 2 m + l (a) Find P2(:r) the second degree Taylor polynomial at a = 0. l l
‘ .—'
" .L = ~\ l. ~ H '
l‘hO = :32; UN): :1 P3 (XE \ 7} >4 2; X
\H :— .l. ’12 “J 'X X
‘l‘ (X) Q; (ﬁll) L.‘ 2 l 4 7i " g
(b) Find R203) the remainder for the second degree Taylor polynomial
at a = 0. 5
l 2
at "(3) 3 {i <% H 3
R m = ”6““) “1 ~——— ’4
a 3‘. 3 A 6. (12 points) For
f (93) = 008W) (a) Write out the ﬁrst 4 non—zero terms of the Maelaurin series for f (50 2 L\ £7 93
X X X X
‘: \“~,~"’ .—___ +_____ . A.
C05 X 9* Li I. (cal i.
L‘ g l 2 36
Comm=x~i+i~i +1: ,
a a. J; (3i (3) \_
(b) Find P4(:c), the fourth degree Maclaurin polynomial for f (1r)
, “r
...  _ >4.
‘ COS(:L‘2) — 1 + £23
(0) Compute EJT
You must show your work in order to receive credit.
l . x“
r' ' . ‘
x4ﬁl+£rﬁ..,.~x+ﬁ 3
.: gm a W G! 11 _ ,._ :
7i\§C )«El , Li L 7. (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. a» ea H To n=l b) If 2 an converges then lim on = 0 (a F n—>oo
n=1 c) If Z(—1)"an diverges then: an diverges. T 6‘)
n=1 n=l 00
If 0 s on < bn and Z on converges, then d) — To) 00
E bn converges.
n21 e) If 2 an diverges then lim an 75 0 T ® n21 ...
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 Spring '07
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