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AMQWW EXAM 2
MATH 2153 SECTION 2, FALL 2010 INSTRUCTOR: WEIPING LI Print Name and Student # SHOW WORK FOR CREDIT 1!! SHOW WORK FOR CREDIT H! (1) (Spas) The curve 1; = 4 + 22,0 S w 5 1 is rotated about the yaxis. Find the area of the resulting surface. I 3 = j 271x :13 i d3: 1+ ($5?)z AX B= “Xi
o d)’
v i =.5H4x* Ax a;th
‘8 217xJH—4x2 4X 2‘ clu=8xdx ,2, = 5,271“ 9; :3: éMS/zfs : 2: (55/2... 4 2 l 1
(2) (8pts) (3.) Find the limit of the sequence an =  lvl X . J"
X—ﬁw X X500 5X:
“M an : O 9
W'>oo
(8pts) (b) Find the limit of (Ln = by using the squeeze law.
,L ~ .L <+
O 5 Q 5 7. ‘rwv 2 :— O
1 n ' n ’ n l
\m m“! :0 1m ago. 17%st 2w W
k—wo ’ 1 mm»
W ‘~1SS('VIZI4 5‘  9/—
_I_ 5 M (.L‘ (‘mtﬁl=o f/rﬁwaéféw. 91133520 h: “ I17— hﬁw WEIPING LI (3) (8pts) Determine whether the series 2‘” (——1)“1“Tn is convergent or divergent. n=3
‘ . " Inn
A'famd‘mj genes [66; b": n I LX' ’"X
x r; x
(n )9" decreasfvj; frx) ‘ a ﬁx) X;
‘ :; ’“IMX Hana +02 H25) ’9" I‘S decreasinj x2
‘3“ > bitH . ﬁ" X > e.
' ~ ‘ '“X .. .‘ '1; ,
(1) JLMMf(x) *1/34“: X Fig/loo l ’ D E" : o ' Hence 'H‘L Series 15 Convetjed'.
0O (4) Show the following series are convergent and ﬁnd their sums.
mmmz%%r ac am! e boo en" 8 .
:Z shE'E'QT) I Y:~.<[. 14!
H1; 4‘ 4a
__ e  4M
“1—3: ' 46
4 ,
'+ Geomet’t‘c Sal/Q95 wF‘W 1Y[= (I I is v Ctnvefjw+ . .. 3 3
(11) (1013175) 220:1(m ‘ (n+1)(n+2))' _. .. .2— _§.
§'—a'"tz z. 3 3 ; Dir}.
3 v... + r ‘1“ < . 34:.
gramm=.z as :55‘4 '3 5
g 3  i a; _. ~—
35 :7; "" 3"!— + g (L $.1— [Z 41..
F 3 3 _.
S" =a,+az + +61” " 7—; (M00117)
3 ‘ 3 .. 3..
Lil"! S " "— v "M p ("+2 2.
‘1 z “7&01D ) CALCULUS III  3 (5) (Spts) Determine whether the series is convergent or divergent. You must provide reason for your claims in order to receive any credit. 00 2 —1
(a) 211:1 3+1 ' lim a“ =1 [5M 11.),» moo V‘ﬂ h—no "HA .L (6) Determine whether the series converges or diverges. Must provide the right test to have any credit. 2 L
(i) (5pts)z:;°=1ne—"  k, x1 J5 vx L
' v = (‘m X 8 dx M = x
11?er E51, X 5‘ 4X J?“ l L dqzzxdx
‘3 ' b —u . ’ ' b (
_. I'M Q Q :: (m e _ E. z, .6...
M L [0904 t 2 19m 2 7 3 1
—l4
2 v1 Q .3 Comij “17
14::
(ii) (513%) 230:2 {Kimmy Comfdﬂ‘fq: b" = M4 : n: ’
Rd, ‘ an a! [’11: :[f‘W'l ——/'TF :1 >0 w‘oo L,” hvno n‘ﬁc4/n4 “*7 179 9Q 60 1 it + g
hZi En 2’: ’15 j / , n,
“a , 3.
Hence I an '5 deaf '
n=< fvseriés . g WEIPING LI (7) (15pts) Find the sum of the series 22:1 ((54%; Ah'wnatr‘uj 3068.4 error esff‘m'o—C With the error of less than 10‘2. (Rn, 4 bVHl '2 1
So bn+[$("1+7)z < '0 ’0 < (“7) ,
n: 4 ‘
Thus S :._ Q.+ a2. + + 04L
4.  I ("3 J...
(') ——~ + f" + z (8) (15pts) Determine whether the series is absolutely convergent, conditional convergent, or divergent. h +0 ’ 2n
(1)2113}? R “(7‘ “ﬁt . {I‘M an” :3 iw1 ( ’ «v
“7‘0" 5‘” WM» 2" "I
= lim 1:1... :. be > I
Irma z ‘/n
i A]:
(11) 220:1(1+;11)—" hr‘ J.
oo't Tés‘t 1m, Jmnl =l;wnoo[('+ n ]
hf>oe ‘ ‘q ’ ’11") ‘
:Z— l I :— ‘. ___‘
“L: (I + n ) I’Lmoo[ n ) MEN}?
:_ f," < I Hence W Sane; 75 (iii) 220:1(—1)”ﬁ “it'ij saws 1—951— E l ‘
I ' ' w ..L— : 'W) "a (Mm
Ln:J—‘:~‘ b":"J‘l‘q—_= WT bx!) Prim 9:1,)!“ +
“W mi“ ‘5 CW5?“ , M" Z (6')an / :5, 72’ 4W5” CALCULUS III 5 (9) (Bonus 10pts) A sequence {an} is given by a1 = V5, an+1 = V3 + an.
(i) Show that the sequence {an} is bounded above by 3. 133—qu, {MA/“0+;9ﬂ inucipie, Q‘L'Jg‘ < 3 .
Surfose 0”. < 5 ‘ Hence Q“H;}5+qn < 15+3 =f—6_ <5415;
Theefme an<5 1can n. (ii) Show that the sequence {an} is increasing by induction. a.=J'5‘, a; 45:3 > 5w ‘53:“:
“W {MA/Raﬁﬁn 73"?"53’9‘6 J arrl < an.
5 if. an’ < 3 + “V! . QMA <.15+0\n‘ ya
Lnyiui'Hm a, 4374“ < Jsﬁh ' "H (iii) Find the limit unlwm an_ I ~ I’VL
fa } '$ L and Menﬁf‘onical9 IMCMQSJ .
n I d u 53
To, Hence In ((1975:; H (m an = L emf“ m m" a /' 4
. ,= 7" 'm .4
. ‘ :9 5
“w 1;me + w W
00 .
A :1 JSI‘L ’ 3
. ’ L, ; ...
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