Mathematics 3100 - Test Three
1. Show that the following series can be rewritten as telecoping series, use this to prove
they converge and evaluate their sum.
(a)
1
1
.
2 1
n
4
(b)
1
(1)n1
.
n(n + 2)
Mathematics 3100 - Homework IX
1. Find the radius of convergence for the following power series.
Solution:
(a)
xn
. We can use either the root test and ratio test to nd the radius of
n n
1 2
converge
Mathematics 3100 - Homework VIII
If you have noticed any mistake or typos, please let me know.
1. Test each of the following series for convergence.
Solution:
n
n
n
1
(a)
. Note that 2
2 = 3/2 . Then
Mathematics 3100 - Homework I
1. For each of the cfw_an below, tell if the sequence cfw_an is increasing (strictly), decreasing(strictly, or neither, show reasoning.
(a) 1
(b)
(1)n1
1 1
+ +
.
2 3
n
Mathematics 3100 - Homework III
2n 1
= 2 directly from the denition of limit.
n n + 2
1. Show that lim
Proof. Given > 0,
2n 1 2(n + 2)
5
2n 1
2 =
=
<
n+2
n+2
n+2
if n >
5
2.
2. Prove the sequence an
Mathematics 3100 - Homework II
1. Let cfw_an and cfw_bn be increasing; are the following increasing? Prove or counterexample.
(i) cfw_an + bn (ii) cfw_an bn .
Solution: cfw_an + bn is increasing s
Mathematics 3100 - Homework IV
1. Prove Leibnizs famous result
1 1
(1)n
= lim 1 + +
.
4 n
3 5
2n + 1
(a) Derive a formula
1
+ en (u).
1 + u2
1 u2 + u4 + (1)n u2n =
(b) Integrate every term from u = 0
Mathematics 3100 - Homework V
1. Prove the Product theorem: an L, an M = an bn LM by following through
the ideas given in the book.
Proof.
|an bn LM | = |(an L)bn + L(bn M )| |an L|bn | + |L|bn M |.
I
Mathematics 3100 - Homework VI
1. Given a set of nested intervals [an , bn ], n = 0, 1, 2, , for which bn an 0, prove
the sequence a0 , b0 , a1 , b1 , a2 , b2 , , an , bn , converges.
Proof. Let the s
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Mathematics 3100 - Test One
1. Assume the geometric formula
1 + a + a2 + + an =
1 an+1
1a
(a) Derive a formula
1 + u2 + u4 + + u2n =
1
en (u).
1 u2
and nd en (u) explicitly.
Mathematics 3100 - Test One
I will begin with some general remarks about the test. I consider the test is dicult
but not as dicult as most of you think. Two problems (3 and 4) have been assigned as
ho
Mathematics 3100 - Homework VII
1. Show that the following series can be rewritten as telescoping series, and use this to
prove they converge and evaluate their sum.
(a)
1
21
n=2 n
(b)
(1)n1
n=1 n(n +