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)
Solution: a). We can rst factor n2 1/4 = (n 1/2)(n + 1/
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
convergence.
2n+1 n + 1
1
an
= n
lim
= n 2 1 + = 2.
lim
lim
n n
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 the comparison theorem im2+1
n +1
n
n
1 n
plies this s
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
n
.
n+1
n
sin2 k.
(c)
k=1
n
(d)
sin k.
k=1
1
(e) sin .
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 =
1
1
1
+
+ +
has a limit.
n+1 n+2
n+n
Proof. You just
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 since
an an+1
and bn bn+1
implies an + bn an+1 + bn+1 .
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 to u = 1. Show that the integral of the error
term tend
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 |.
In Problem 3.4, we have shown that every convergent sequ
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 sequence be cfw_cn . Then we can write
cfw_cn = cfw_an
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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.
(b) By integrating from 0 to
1
2
of the above for both
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
homework. The other two problems are very close to the 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 + 2)
(c)
1
, k integer > 0.
n=1 n(n + k)
Proof. When a s