Math 252
Exam 3 Key
Instructions
1. Do NOT write your answers on these sheets. Nothing written on the test papers will be graded.
2. Please begin each section of questions on a new sheet of paper.
3. Please do not write answers side by side.
4. Please do not staple your test papers together.
5. Limited credit will be given for incomplete or incorrect justification.
6. All numeric answers must be exact: numeric approximations are not acceptable.
Questions
1. Integrals (6 each)
Z ∞
1
(a)
dx
1 + x2
0
Z
∞
1
dx
1
+
x2
Z0 a
1
lim
dx
a→∞ 0 1 + x2
a
lim arctan x|0
a→∞
lim arctan a − arctan 0
a→∞
Z
(b)
0
1
=
=
=
=
π
.
2
1
1
−
dx
x x−1
1
Z
Z
1
1
−
dx +
x x−1
Z
1
1
−
dx + lim−
x x−1
b→1
Z
0
Z
lim+
a→0
a
1/2
1/2
1/2
0
1
1/2
b
1/2
1
1
−
dx
x x−1
=
1
1
−
dx
x x−1
=
1
1
−
dx
x x−1
=
b
lim ln |x| − ln |x − 1||a + lim− ln |x| − ln |x − 1||1/2 =
b→1
1/2
x x b
lim ln + lim− ln =
x − 1 a
x − 1 1/2
a→0+
b→1
a + lim ln b − ln | − 1| =
lim+ ln | − 1| − ln a−1
b − 1
a→0
b→1−
a→0+
1
∞.
Exam 2
2
2. Sequences (4 each)
(a) an = ln n − ln(n + 1)
lim ln n − ln(n + 1)
n
lim ln
n→∞
n+1
n
ln lim
n→∞ n + 1
n
ln lim
n→∞ n + 1
ln lim 1
n→∞
∞−∞
=
=
=
∞/∞
=
n→∞
(b) bn =
continuity
L’Hôpital’s Rule
=
ln 1
=
0.
32n +n
3n
32n + n
n→∞
3n
2n
3
n
lim
+ n
n→∞ 3n
3
n
lim 3n + n
n→∞
3
1
lim 3n +
n→∞
ln 3(3n )
lim
∞/∞
=
=
∞/∞
=
=
L’Hôpital’s Rule
∞.
(c) cn = (ln n)1/n
lim (ln n)1/n
n→∞
e
e
lim eln(ln n)/n
n→∞
ln(ln n)
limn→∞
n
ln(ln n)
limn→∞
n
e(
limn→∞ ln1n
)
∞0
=
=
=
∞/∞
=
=
continuity
L’Hôpital’s Rule
1.
Exam 2
3
3. Series (4 each)
For the two series determine whether they converge or diverge.
(a) Write the definition of an infinite series. You may use the words or the symbols.
The value of an infinite series is the limit of the sequence of the partial sums. Formally
∞
X
i=1
(b)
(c)
ai = lim
n→∞
n
X
ai .
i=1
∞
X
3
5n
n=1
This is geometry with a = 3/5 and r = 1/5 < 1. Thus this converges.
∞
X
32n + n
n=1
3n
Based on the limit above, this diverges by the nth term divergence test.