MATH 1220-6
Fall 2003
Midterm exam III
Student Name:
Student ID Number:
Course Abbreviation and Number:
Course Title:
Instructor:
Math 1220
Calculus II
Vladimir Vinogradov
Date of Exam:
Time Period:
Duration of Exam:
Number of Exam Pages:
(including this cover sheet)
Exam Type:
Additional Materials Allowed:
November 20, 2003
Start time: 7:00 pm
1 hours
10
Closed Book
Calculator
QUESTION VALUE SCORE
1
20
2
20
3
20
4
20
5
20
TOTAL
100
End Time: 8:00 pm
1. (20 points) Does the series converge or diverge? Give reasons.
∞
X
n!
a)
en
n=1
ANSWER:
2
b)
∞
X
(−1)m
m=1
esin(mπ)
m2
ANSWER:
3
2. (20 points) Use the Integral Test to decide the convergence or divergence of the following
series:
∞
X
k2
k e− 2 .
n=1
ANSWER:
4
3. (20 points) Find the sum of
∞
X
2n
(n + 1) n .
3
n=0
ANSWER:
5
4. (20 points) What is the interval of convergence of the power series? Show your work.
a)
∞
X
(3x + 1)k
k=1
k 2k
ANSWER:
6
b)
∞
X
(−1)k (x − 2)k
k=1
k2
ANSWER:
7
5. (20 points) Find the Maclaurin series for the function.
a) f (x) = e2x − 1 − 2x
ANSWER:
8
b) g(x) =
1
2 + 3x
ANSWER:
9
Useful formulae
loga x =
ln x
,
ln a
ax = ex ln a ,
loga xn = n loga x,
ab ac = ab+c ,
(xα )0 = αxα−1 ,
(ax )0 = ax ln a,
Z
1
(ln x) = ,
x
0
x
e =
∞
X
xk
k=0
p
ln xdx = x ln x − x + C
k!
(1 + x) = 1 +
=1+x+
∞ µ ¶
X
p
k=1
k
xk
x2 x3
+
2!
3!
|x| < 1,
µ ¶
p
p(p − 1)(p − 2)(p − 3) . . . (p − k + 1)
where
=
k
k!
∞
X
1
=
xk = 1 + x + x2 + x3 . . .
1 − x k=0
|x| < 1
∞
X
x
=
xk = x + x2 + x3 + x4 . . .
1 − x k=1
ln(1 + x) =
|x| < 1
∞
X
xk
x2 x3 x4
(−1)k+1
=x−
+
−
...
k
2
3
4
k=1
10
|x| < 1