Exam 3
CalculusII, MA 112
l(.s 7
t-
N"*"
Exam 3 (100 points)
May 11,2004
Box-
You may use your calculator but not maple.
(18 pts) 1. Determinethe limit if the limit exists. If the limit doesnot exist, write DNE.
xrr
(15 pts) 2.
g
Answer the following questions about the r;eries
1
1
n+r)
@q
lC
+
(a) The limit of the terms of the seriesis
L
is (givea """"i*l
(b)rhe 10thpartialsum,s16,
\
(you may usethe fact that
L@Tt
1
3
1
Q
-l)*(f
i\r
' + (l^-'/')=
(c) The n'h partial sum, sn, is (give a simplifiedexpression) | - L ^
4+l
(d) The limit of the partial sumsis
I
3
(e) The infinite seriesis - circleeither "€@
Z
(f) If the answerto (e) is convergent,then the valueof the infinite sum is
divergent.
I
(11 pts) 3. What is the HarmonicSeries?Is the Harmonicseriesconvergentor divergent?
-y3.
-L
11
fi=t
T)rta^S*
I
-h
=fr
( 2 1pts)a. To the lefb of the series, write C if the series is convergent or D if the series is divergent.
,rb
,il
d
To the right of the series, give your reason - you need not do the details - simply put the
method (or theorem) which you would use to justify your answer (for example, the series
is geometric, the integral test, comparison test, p- series test, terms do not convergeto
zero...). However, you need to give the information required in the test: for example, the
geometricserieswith r: |, the comparisontest with ar,: ... and bn: ..., thep seriestest
with p :...
o
C
o
r\a
a ) )
'
u
h
o
p-v-ahr'F';tt
6
tt
-
'u
n:\
T'a/;^
C ori4
' /-t
n"/n
""tlLr , * , 1
n=1
oo
n
U
^" \' \' -l Z n5*nr
3
a
^fr[4ta'*'->
?+ o
flr
n:l
-7
c
o
d' )t.(-/
n
lJ
a
n'
- - l
.)i (3)"
'tAtu*
o
t-lzr,*f A
( 1- : ) "
u e
GuJr=?r
D"Ery ClKdn
ffi/
ov
nrre
fu
,(,u
h
L
o
o
. , \/ \ o
4
7
).-
o
'
lrnx
1+n3
oo
/
1\n
(10 pts) b. Find the valueof the convergentinfinite seriesfz-,/ +.
3n
Note that the seriesstarts with
n:l
^
I L
-'l
_
L .
%
=s
a
,/1/ai
c / ) ":s ! / - ' l l n =
Y
*
I
C,l-t)
-L:
t-ril
tq'=L-t =-J
n t , v I ' Y
+t/s
=)I
(10 pts) 6. Use integration by parts to find
V=L
d/"=-/x
'"-"a''
|
A.r=€kk
/ --Cr
e-KJv
*
e
K
f
C
x
{
.
n
o
{
r
=
J
- ( - - Kf c
=-9e
to n"a
usepartialfractiondecomposition
z K 1 \
-fiv+r\
----
r , l
(
| ffih,
x* l
+c
+
/^(x+rl
trl
,',$o*- h
(15 pts) Z. Determine the convergenceor divergenceof the improper integral
[* "-"d"r.
J o
A
qo
( e-'Klr='k-*
lr'vd
r
l
fre-qk
r
-&
,L
LO
€
-
ko*
-L
-t*
L
/_-
7-'
/L-b
,',crr*
lim re-'.
UseL'H6pital's rule to find t+oo
tH'V/)
,Lb e x
X
-
ex
LaP
La
)<-5 r'
-L =o
lv-
J
lim r" : 1.
UseL'H6pital'srule to showlt+u
l"(\ = * . h r
l^
=trL V^t)
1--r'3 L
kro
li
'.
l
"V*
(
/
l*
K-+o
h(
x
F-a
0
,L9
n9o
,tAx =
/
=
L
X
qx-
-(
=
/-
d s/
N-0
=0