Midterm 3 Practice
Exam Date: November 26, 2014
You will have 50 minutes to complete midterm 3 plus an additional 30 minutes to work on
2-3 problems that are like those from midterm 2. On this exam you can bring one 3”x 5”
notecard with any information you find useful for the exam. Please note: Nothing bigger
will be allowed - it is one 3” x 5” notecard!
1. Complete the following statements:
(a) The sequence {an } is said to converge to L if for each positive number ε ...
(b) The infinite series
∞
X
ak converges and has sum S if ...
k=1
(c) A power series in x has the form ..
(d) A Maclaurin series is ...
1
2. True or False. If false, correct the statement.
(a) If
∞
X
|un | converges, then
n=1
∞
X
un converges
n=1
(b) If lim an = 0, then
n→∞
∞
X
an converges.
n=1
(c) For a convergent alternating series,
∞
X
(−1)n+1 an , the tightest bound on the error
n=1
made byZusing the sum Sn of the first n terms to approximate the sum S of the
∞
a(x)dx.
series is
x
2
3. Does the given sequence converge or diverge, and if it converges, find lim an
n→∞
(a) an =
(b) an =
1+
4
n
n
n!
3n
sin2 n
(c) an = √
n
3
4. Determine if the given series converges or diverges and if it converges, find its sum.
k
∞ X
1
(a)
ln 2
k=1
(b)
∞ X
1
1
−
k k+1
k=1
(c)
∞
X
k=0
−2k
e
=
(Hint: write down the nth term in the sequence of partial sums)
k
∞ X
1
k=0
e2
4
5. Indicate whether the given series converges or diverges, give a reason for your conclusion.
(a)
∞
X
n=1
(b)
(c)
∞
X
1
(−1)n+1 √
3
n
n=1
∞
X
n2
n=1
(d)
n
1 + n2
∞
X
n=2
n!
1
k ln k
5
6. Is the given series absolutely convergent, conditionally convergent or divergent?
(a)
∞
X
n=1
(b)
(c)
(−1)n+1
n−1
n
∞
X
1
(−1)n+1 p
n(n + 1)
n=1
∞
X
sin n
(−1)n √
n n
n=1
6
7. Find the convergence set for the given power series.
(a)
∞
X
n=0
(b)
xn
n3 + 1
∞
X
3n x3n
n=0
(3n)!
(c) (x + 3) − 2(x + 3)2 + 3(x + 3)3 − 4(x + 3)4 + ...
7
8. Given
1
= 1 − x + x2 − x3 + x4 − ..., |x| < 1.
1+x
Find a power series that represents
1
.
(1 + x)2
9. Find the first 5 terms of the Taylor series for ex around the point x = 2.
8
10. Find the first three nonzero terms of the Maclaurin series for:
f (x) = e−x sin x
11. Determine how large n must be so that using the nth partial sum to approximate the
∞
X
k
series
gives an error of no more than 0.000005.
k2
e
k=1
9