Infinite Series
Directions: Find the sum of the series.
95.
!
tan(3n)
107.
n=1
96.
n=1
Directions: Answer the questions.
!
4(0.3)n1
&
&
108. &
If
an and
bn are both divergent, is
(an + bn ) necessarily divergent?
Directions: Determine (a) whether cfw_an
164.
!
e1/n
n=1
165.
(b)
n7
(c)
nn
Directions: Find the radius of convergence and
interval of convergence of the series.
166.
n=3
167.
xn+11
n
!
(1)n1 xn+4
n3
!
n xn+5
!
xn+8
n=1
8n!
!
(x 10)n
173.
!
n=2
174.
!
n=1
n=0
Representations of Functions as
Powe
65.
!
x 7 x + a dx
66.
!
12 dx
x+x x
Infinite Sequences
Directions: Determine whether the sequence
converges or diverges. If it converges, find the
limit.
2n+6
7n
"
#
3n
80. an = tan
4 + 12n
79. an =
Improper Integrals
Directions: Evaluate.
!
4 dx
67.
(5
1. Let the function be 3 =f (1+ LIZ) , where f has continuous two second partial
525
:terivatives , calculate .
615)?
2. Let the points (1 ,3,2),(2,1,3) and (-1, -1,2) are all on the plane, nd the equation
of the plane it.
Directions: Evaluate the indefinite integral as a
power series and determine the radius of
convergence.
185.
!
ln(1 t)
dt
t
186.
!
x arctan x
dx
x3
187.
!
arctan(x2 ) dx
195. Find the Taylor series for f centered at 2
given that f (x) = x4 6x2 + 1.
196. F
3. Calculate the curve integral LEW, where the curve Lis the positive
direction of the circle (A' DZ + y2 = 4.
4. Calculate the surface integral 12M+12yadr+fzdmji . where surface 2 is
E
the part ofthe cylinderx2 +y2 =1 between the plane Z= Oand Z = 2 , ta
Suppose the hemisphere Q be 12 +y2 +Z'2 E 4 , 32 O, and the density ofth:
hemisphere Q is given by PU, y, Z) = z.
( 1) Find the mass of the hemisphere;
(2) Find the center of mass of the hemisphere.
The curve y=ax^2+bx+c passes through the point A(2,0) abd B(3,20).The slope of the normal at A is 1 and
the gradient of the tangent at B is 9.Find the values of
a,b and c.
139.
!
(1)n
n=1
140.
!
sin 4n
!
Directions: For each of the following series, is
the Ratio Test conclusive or inconclusive?
!
2
154. (a)
n3
n=1
7n
n=1
141.
e1/n
n7
!
n
(b)
3n
n=3
12n
(n + 1)82n+1
n=1
!
(4)n1
n
n=2
!
4 n
(d)
1 + n2
n=2
!
n8 5n
142.
(1)n+1
1. Find the solution of the differential equation
J"-4y'+4y=0, KO) =0, J"(0) =1-
2. Let the power series be 2 I , find the sum function ofthe series and point o t
:FI
the convergence domain.
249. Find the area inside the larger loop and
outside
the smaller loop of the limacon
3
+ cos .
r=
2
(a) Find the volume if b = 7, a = 5, and
h = 4.
(b) What are the shape and volume if
a = b? (Express your answer in
terms of b and h.)
250. Find the pola
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