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SECOND SAMPLE FINAL FOR MATH 2J
This sample 2J final exam is meant for practice only. Length and content of the
actual final may vary, at the discretion of the instructor.
Try to complete the test in 1 hour and 50 minutes, without consulting any book or
notes, and without using a calculator.
1
2
PROBLEM 1 (True/False Questions)
If 0 ≤ an ≤ bn and
∞
P
bn converges, then
n=1
true
If 0 ≤ an ≤ bn and
n=0
2n
n!
an converges.
n=1
f alse
∞
P
∞
P
bn diverges, then
n=1
∞
P
∞
P
an diverges.
n=1
true
f alse
true
f alse
= e2 .
1x + 2x + 3x + . . . is a power series.
true
f alse
0
If P3 (x) = 4 + 2x − x2 is the 2nd -degree Taylor polynomial of a function f at a = 0, then f (0) = 2.
true
f alse
If {an } is decreasing and an > 4 for all n, then {an } is convergent.
true
f alse
If an > 0 and
∞
P
an converges, then
n=1
true
∞
P
(−1)n an converges absolutely.
n=0
f alse
If 0 < α < 1, then limn→∞ αn = 0.
true
f alse
3
PROBLEM 2
Determine whether the sequence is convergent or divergent.
If it is convergent, find its limit.
√
{−5 + (0.9)n }
diverges
converges to: . . . . . .
{4 + (−1)n }
diverges
converges to: . . . . . .
n3 + 2 −
n
√
n3 − 2
n2 +2n−1
n−2
o
diverges
diverges
converges to: . . . . . .
converges to: . . . . . .
4
PROBLEM 3
Determine whether the following series are convergent or divergent.
Write the name of the tests you use.
∞
P
n=0
n2
n3 +1
converges
diverges
n2
3n
converges
diverges
converges
diverges
converges
diverges
∞
P
n=0
∞ P
n=0
n+2
3n+1
n
∞ q
P
n−1
n=1
n
5
PROBLEM 4
Consider the series
∞
X
(−1)n √
n=0
1
.
n+2
(1) Is the series convergent? (Motivate your answer.)
(2) Is the series absolutely convergent? (Motivate your answer.)
6
PROBLEM 5
Test the series
∞
X
1
n(ln
n)2
n=2
for convergence or divergence.
(Motivate your answer.)
7
PROBLEM 6
The following series are convergent. Find their sum.
∞
X
2n+2
3n
n=0
∞ X
2
3
+ n
3n
2
n=0
8
PROBLEM 7
Consider the telescoping series:


∞
X
 1
1 


 3n + 2 − 3n + 5  .
n=1 |
{z
}
an
(1) Write down the first 4 partial sums:
s1 = a1 = . . .
s2 = a1 + a2 = . . .
s3 = a1 + a2 + a3 = . . .
s4 = a1 + a2 + a3 + a4 = . . .
(2) Find the sum of the series.
9
PROBLEM 8
a Find the limit of the sequence
b Test the series
∞
P
1+
n=1
(Motivate your answer.)
3 n
n
1+
3 n
.
n
for convergence or divergence.
10
PROBLEM 9
Consider the series
∞
X
2n (x − 2)n
√
.
n
n=0
(1) Find the radius of convergence.
(2) Find the interval of convergence. Don’t forget to check the end points!
11
PROBLEM 10
Suppose that
∞
X
(−1)n xn
n 3n
n=1
is the power series representation for a function f (x) around a = 0.
a) Find f 0 (x).
b) Verify that f (x) and f 0 (x) have the same radius of convergence.
c) Do f (x) and f 0 (x) have the same interval of convergence?
Hint: check the end points.
12
PROBLEM 11
Find the Taylor series of ex centered at a = 2 .
Find the Taylor series of e−x centered at a = 0 .
Find the Talylor series of x2 e−x centered at a = 0 .
Find the Talylor series of sinh (x) =
ex − e−x
centered at a = 0 .
2
13
PROBLEM 12
a) For all n from 0 to 5, calculate the nth -degree Maclaurin polynomial Tn (x)
for the function
f (x) = 2x3 − 6x + 2.
f (n) (x)
f (n) (0)
Tn (x)
n=0
n=1
n=2
n=3
n=4
n=5
(b) Find the Maclaurin series of f .
c) For all n from 0 to 5, and all −2 ≤ x ≤ 2, use Taylor’s inequality to estimate
the remainder Rn (x) of the Maclaurin series.
|Rn (x)| ≤ . . .
n=0
n=1
n=2
n=3
n=4
n=5
14
PROBLEM 13
Find the Maclaurin series for the function
1
.
(2 + x)2
15
ADDITIONAL PROBLEMS FOR PRACTICE
16
PROBLEM A
The terms of a series are defined recursively by the equations
7n + 1
a1 = 5,
an+1 =
an for all n > 1.
6n + 3
(1) Find the first four terms of the sequence.
a1 = . . .
a2 = . . .
a3 = . . .
a4 = . . .
(2) Find the nth term of the sequence.
an = . . .
(3) Determine whether the series
∞
P
n=1
an converges or diverges.
17
PROBLEM B
Find all the values of p for which the series
∞
X
p
n 1 + n2
n=1
converges.