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Unformatted text preview: Math 122 Test 3 i i November 22, 2005 A Total Directions: CJI . Numerical experiments do not count as justification. . No books, notes or or drawing comical pictures of your Chemistry 111 structor. You may use a calculator to do routine arithmetic computa—
tions. 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. 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 l can’t follow what you are doing7 you will not get much credit. . You may leave as soon as you are finished3 but once you leave the exam" you may not make any changes to your exam. . The exam has 7 questions and a total of 100 points. Happy Thanksgiving 1. (10 points) Find the limit of the following sequence: V712 +n— V712 — n}ocﬁ2 r.” \ 2. (10 points) For Determine Whether the series con'vergeS or diverges. ﬁnd its sum. «w \ﬂ; M23}
’2’” ":5 MM w {)0 Z n:0 1 + 3“
5n v» ,
xi w
,5 '3 NR
\ta: ) If it converges, 3. (20 points) For each of the following series7 determine if it converges or
diverges. Fk'u' each test you use, you must name the test, perform the
test, and state the conclusion you reached from that test. 00 n
TL
‘1
(( > (n + 1)
t m
WWW (“a = Q Y"; W \k W w NJ,“ is Wm 35 “M kkkkk V m )f h J N “is; {in Kﬁx“ﬁ C: ‘1 it; “:1; E OO 1
(C) 2 3712 + 4n + 5 77:} km GET” “Ifilgtmgz e to, Kim \J 4. (20 points) Determine if the following series converge almolutely, C(ny
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 ‘ .i t r I ,t. \ M x (a) Z<~1>ﬂ arc/tan rz/ Lag W; _ {ﬂag (I) y
“:1 1 + n?
K ,. ,m“ l” (“3M ,
min“? MW 4 “ia 7;» «cm:
KQ ‘ AL AAAA ,. a m A m “T?” W LEW ’15???
to w 1 ﬂ  thug: ,3) mg 35;? ﬂ { ““““ _ 1., « ~ " VYMN \‘ V ' ' 3x3 «m MAN‘er KM
Tthm . K MW 7 ""““ WWW
l: at (M six«9 ET h «13,535 “a
a? W
00 1n n n
1 n. . M“
(c) 2H)” (ﬂ) L t J , m A;
,9 n
n—H
£2 {:2 m a” @525,» “a” ' W W I M ﬂ
@ ﬁ 7N
y»: i Erna!“ M g CV1, RR
[ﬁx > i .mvw 'W L 2 an ‘I‘M¢/V“ % “an” 39%“: WNW Xx ‘ N M)! N ‘9 Wngwa W 5, 3m :5; C; 5.: J (15 points) Consider the power series: 00 (1; .i— 2)” “2%” 7L \/ 1'1 72, (a) Where is the power series centered? gown (13) Find the radius of convergence. 1 , . WWW g M “vii “i (:53 x ‘ iii“ _
i Mt: \ K rwja‘\ 41K \ “ii. {3:3
(Ex; ’2'"Z”‘”T”""§x ;. ,,
\N is \ 3 \ties'tw * (m an) (c) Find the interval of convergence. [Hintz Check the endpoints.] :32 m
‘9 L) X
\XCJ
C W ‘ k If,» ........ \ V y \ (Fm ‘ Mi "I" ’35
M) wumww fnﬂmw.
K wﬁ invmt
he“ RVEan 3; ﬁve“
E a
i N
gum ._ N M” (cm ‘i 3 , ‘‘‘‘ m x M ........... W. ,3“ V” W  5
4%.MVTW mmwwimw
\ \ "j: Mn?”
5
i r 6. (15 points) Given the Maclaurin series for f (:13) is \ ngmB Match each of the following functions with the correct Maclaurin series: \1 a) 1 + 1:2 + 3’; + £339 +
“2;? 2 ‘ ,3
CM b)1+2:1:+Z—17§——r§'7’y+ , 2.2 » ,.3
jg?) C)1+2m+g.§_+l7m +
.2
m H 2 3
.1... f(:1:)(sin d) 1 + '21: + 1—55— ﬂ of 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. 00 “W‘in
a) If 2 anar" converges at = “3, it converges at a: :: F
7L:1 NW
00 “4
If a 1’ 1 " conver es for :1: = ' and “A
— . n2
:2: = «~1 then it must converge for CE 2 —1.5.
If the sequence {am} converges then the C) an V T _ F
sequence ~77 converges to 0. / DC 00
If the series an converges and the series 2 b” con— n=1 'rL::l f
d) 00 T I F
verges (an and b,” nonnegatrve), then the serles 2mm“) n21
converges. 00 00 e) If 2 an diverges, then 2 km diverges. T F
11:1 nzl ...
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 Butler
 swers., Verges, Kim \J

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