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Problems for lecture 22
April 17, 2014
1. Recall that a series of numbers
ak is convergent if and only if the series is Cauchy (i.e., for
every > 0, we can nd an N so that |am+1 + am+2 + + an | < for all n > m N ).
We want to have a similar result for a s
Problems for lecture 25
April 29, 2014
1. We know that power the series k=0 xk converges pointwise on (1, 1) to f (x) = 1/(1 x).
As a consequence of the theorems on power series, we can say that this series converges
absolutely on (1, 1) and uniformly on
Problems for lecture 22
April 24, 2014
This homework set is about the summation by parts formula
n
n
ak .bk = sn bn+1 sm bm+1 +
sk (bk bk+1 )
k=m+1
k=m+1
where sn = a0 + a1 + a2 + + an . You can see this formula in Exercise 2.7.12 on page 68 in the
book.
Problems for Lecture 1
January 14, 2014
1. Using the -denition of a limit, prove that
lim
3n + 1
3
= .
2n + 5
2
2. Using the -denition of a limit, prove that
lim
10
= 0.
(n + 1)(n + 2)(n + 3)
3. We learned in class that if a sequence is convergent then it
Problems for Lecture 3
January 29, 2014
1. Suppose that the sequence an is decreasing and bounded. Show that the sequence an converges. (Do not say that this is a consequence of the Monotone Convergence Theorem, we
are trying to prove this theorem.)
2. We
Problems for Lecture 2
January 16, 2014
1. Let c be a xed real number. Prove that if lim an = a then
lim c.an = c.a.
n
2. Suppose that an is a positive sequence and lim an = a = 0. Prove that
lim an = a.
Hint:
|an a|
|a a|
n
| an a| =
.
an + a
a
3. Cons
Problems for Lecture 4
January 23, 2014
1. Use the Cauchy denition to show that the sequence
1
2n
is a Cauchy sequence. (Do not use the fact that a Cauchy sequence is the same as a convergent
sequence).
2. Do not use the fact that a Cauchy sequence is the
Problems for lecture 6
January 30, 2014
1. We know that a dierent rearrangement the series
(1)n+1
n
n=1
may converge to a dierent limit. In fact given any number c, we can nd a rearrangement
which converges to c. Can you nd a rearrangement of this series
Problems for Lecture 5
January 28, 2014
1. Prove Alternating Series Test which says that if b1 b2 b3 0 and lim bn = 0 then
the series (1)n+1 bn converges.
2. Consider the series
an .
(a) Show that if the series
an converges absolutely then the series
a2 c
Problems for lecture 7
February 4, 2014
1. The denition of limxc g(x) = L says that for every
0 < |x c| < then
|g(x) L| < .
> 0, we can nd > 0 so that if
Use this denition to prove the Squeeze Theorem which says if we have f (x) g(x) h(x),
limxc f (x) = L
Problems for lecture 9
February 11, 2014
1. Show that the a sequence xn converges to x if and only all subsequences xnk converge to x.
2. Prove that if a set A in R is closed and bounded then it is compact.
3. Determine if the following sets are open, clo
Problems for lecture 10
February 13, 2014
1. Decide whether the statements below are true or false. If it is true, provide a short proof . If
it is false, provide a counter example.
(a) If K is compact and F is closed then K F is compact. Hint: you can us
Problems for lecture 7
February 6, 2014
1. Use the and -denition or the sequence xn -denition to show that the functions below are
continuous at given points:
(a) f (x) = 3 x at 8. Hint: you can use the identity a3 b3 = (a b)(a2 + ab + b2 ) with
appropria
Math 4606, Summer 2002: The Intermediate Value Theorem and two proofs; recursion
Page 1 of 2
The Intermediate Value Theorem: If f (x) is continuous on [a, b], and there exists a real number y such
that f (a) < y < f (b), then there exists xo , where a < x
MATH 401 - NOTES
Sequences of functions
Pointwise and Uniform Convergence
Fall 2005
Previously, we have studied sequences of real numbers. Now we discuss
the topic of sequences of real valued functions. A sequence of functions cfw_fn
is a list of functio
Math 341 Lecture #32
6.5: Power Series, Part II
For a power series with radius of convergence R > 0, we have shown that the power series
is continuous on (R, R).
We saw in an example that the interval of convergence could be (R, R], and the question
is th
Summation by Parts
An important technique of calculus is integration by parts:
b
b
a u(x)v'(x)dx = u(b)v(b) u(a)v(a) a u'(x)v(x)dx
This is useful, obviously, when u'(x)v(x) is easier to integrate than
u(x)v'(x), e.g., if u(x) = x and v(x) = ex.
An analog
Problems for lecture 21
April 16, 2014
1. Prove that if fn
f on A then for every > 0, we can nd N N such that |fm (x)fn (x)| <
for all m, n N and all x A.
2. Consider the sequence of functions
1
fn (x) = x1+ 2n1
on [1, 1]. Show that fn (x) converges unifo
Problems for lecture 22
April 10, 2014
1. Given a sequence of dierentiable function fn (x) on [a, b]. A common question is whether
the derivative of a limit is the limit of derivatives (i.e., will we have (limn fn (x) =
limn fn (x)?). This is certainly no
b
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Problems for lecture 12
February 20, 2014
1. A function f : A R is called Lipschitz if there is a number M > 0 so that
f (x) f (y)
<M
xy
for all x, y A, x = y.
(a) Show that if f : A R is Lipschitz then it is uniformly continuous on A.
(b) Consider the fu
Problems for lecture 10
February 19, 2014
1. Let f :A R be a continuous function and K be a compact subset of A. Show that minf (K)
exists.
2. Consider the function
f (x) =
1
.
x2
(a) Show that this function is uniformly continuous on [1, ).
(b) Show that
Problems for lecture 13
February 25, 2014
1. Assume that f : R R and g : R R are dierentiable at a point c A.
(a) Prove that for any k R, the function kf (x) is dierentiable at c and its derivative at c
is kf (c).
(b) Suppose g(c) = 0. Use the fact that g