Math 163, Section
Some problems to practise for Hour Test 2
These problems are intended to help you test and strengthen your conceptual grasp of Chapter 25, Chapter 26, Chapter 27, Chapter 28 and
a tiny bit of linear algebra and there is no guarantee tha

MA 163
Hour Test 2
Spring 2011
Name:
Student ID #:
Instructor: Vadakkumkoor Sandeep Varma
Instructions
1. Write your name on each page. Well, at least on the rst page.
2. There are 5 problems on 8 numbered pages (the last sheet being a rough/extra sheet).

Some Homework Solutions and some Sketches
HW 7, #4 : Read and understand the statement of Corollary of Theorems 1-3 for series
of functions, as well as a statement of the Weierstrass M -test, and use it to prove that the
function f given by :
sin(nx)
x R

Math 163, Sections 43 and 51
Some problems to practise for Hour Test 1
Note : These problems are intended to help you test and strengthen your
conceptual grasp of Chapters 23 and Chapter 24, and there is no guarantee
that the exam problems will be similar

MA 163
Solutions to Hour Test 1
Spring 2012
1 .
(5 points) (a) Dene without referring to any other denition taught in this course so far, i.e.,
using just , N etc. : the absolute convergence of a series of real numbers.
Answer : A series
an of real number

MA 163
Hour Test 1
Spring 2011
Name:
Student ID #:
Instructor: Vadakkumkoor Sandeep Varma
Instructions
1. Write your name on each page. Well, at least on the rst page.
2. There are 5 problems on 8 numbered pages (the last sheet being a rough/extra sheet).

Quiz 1 and Solution
Question : Suppose an > 0 n N, and suppose
that
an converges. Show
n=1
a2n1 converges.
n=1
Proof : Suppose cfw_sn and cfw_sn are the sequences of partial sums of the two
series
a2n1 respectively, so for all n N :
an and
n=1
n=1
n
s n

Math 163, Section 21
Some Linear Algebra problems to practise for the nal examination
Caution :
1. These problems are intended to help you test and strengthen your conceptual grasp of
the material from Linear Algebra, and there is no guarantee that the ex

MA 163
Section 43
Final Exam
Spring 2011
Name:
Instructor: Vadakkumkoor Sandeep Varma
Instructions
1. Write your name on each page. Well, at least on the rst page.
2. There are 7 problems on 14 numbered pages (the last one being an extra sheet).
3. Anythi

Note : Unless otherwise stated, V (strictly speaking, (V, +, ) denotes a
vector space over a eld F (strictly speaking, (F, +, ).
Note : This is the last homework set due(!), and you have a long weekend
this time, hence I am giving this relatively large ho

Homework 24
Due Friday, May 25
Note : Unless otherwise stated, V (strictly speaking, (V, +, ) denotes a vector space over
a eld F .
Homework 24
1. Let V be a nite dimensional vector space, i.e., there exists S V that is nite
and spans V . In class we prov

Homework 23
Due Wednesday, May 23
Note : Unless otherwise stated, V (strictly speaking, (V, +, ) denotes a
vector space over a eld F .
Only four problems, and that too mostly easy ones.
1. First we dene a notion we need for this problem. Let V, W be vecto

Homework 21
Due Friday, May 18
Note : It seems to be that this homework is lighter. And it is made to be so
since you had an hour test today.
1. Informally, this problem is : show that a subspace of a vector space is
naturally a vector space in its own ri

Homework 22
Due Wednesday, May 23
Note : More problems, but most are easier. Unless otherwise mentioned, V is a vector
space over F , i.e., (V, +, ) is a vector space over (F, +, ).
Example of linear independence : Let v V be a single vector. Then cfw_v i

Homework 19
Due Friday, May 11
1. Page 585, #7.
Hint : Suppose you were in (R, +, ) : how could you prove such a thing
- in rough space solve the equations in the high school way (this in
itself doesnt constitute a proof as what you need is not just to el

Homework 20
Due Wednesday, May 16
Note : It is due on the exam day since you do have a weekend in between.
The exercises with references are from Tools of the trade - if you dont have
a copy and have diculty getting it in time, please let me know. Note th

Homework 17
Due Wednesday, May 9
Note : Read the material on pages 563 and 564 and pick up the denitions
of exp, sin and cos on complex numbers. Unfortunately I dont have enough
time to do these - I will review them briey on Monday though - in class in as

Homework 14
Due Friday, May 4
n2
1. Find the radii of convergence of
z ,
n=1
n
sin nz ,
n=1
. For the latter, recall a
n=1
problem I discussed, may be in problem session, that sin n doesnt have a limit;
you will need to replicate the proof.
an z n has rad

Homework 18
Due Wednesday, May 9
Note : This may be slightly heavy especially in conjunction with HW 17, but
I request you to persist and do everything. This is quite important. This is
also why you should have done as much of HW 17 during the weekend :P

Homework 12
Due Wednesday, May 2
1. You know the function g : R R given by g(x) = |x| for all x R is dierentiable at all points of R\cfw_0. In contrast, prove that the function f : C R C
dened by f (z) = |z| for all z C is : (i) Not dierentiable at (1, 1)

Homework 15
Due Wednesday, May 2
1. Find if each of the following series converges, and for convergent series
nd if the convergence is absolute :
(1 + 100i)n
in
1
log n
,
,
.
+i
n!
log n n=1 2n
n
n=1
n=2
2. For each of the following power series show that

Homework 11 Due Wednesday, April 25
1. Page 539, #1 (i), (ii), (iii), (v) (arguments only - you should specify
the set of all possible arguments unlike what Spivak gives at the back
(try not to look at that). You can in fact use a couple of the problems
b

Homework 12
Due Wednesday, April 25
Denitions you will need for this homework
Today we discussed limits and continuity for functions f : B C for a
subset B of C. In Chapter 6, we discussed limits and continuity for functions
A R, for a subset A of R. We w

Homework 13
Due Monday, April 30
1. This is very similar to a problem from HW 12. Only, the domains are
dierent. Suppose A C and B R (so this is the reverse of what
was the case in HW 14). Let f : A B is a real valued function and
g : B C be two functions

Homework 10 Due Friday, April 20
Important Note : I am giving a lot of problems since many of them
are very easy and direct. PLEASE AVOID LOOKING AT THE BACK
OF THE BOOK (or for that manner solution manual) TO DO THESE,
especially if you are not very comf

Homework 9
Due Monday, April 16
1. Review of remainder estimates : in each of the cases below, use Theorem 20.4 to
prove that for all x in the given interval I, the Taylor series of the given function f
at a, evaluated at x, converges to f (x). In other w

Homework 7
Due Wednesday, April 11
Note : We are getting somewhat behind, so I am asking you to read the
statement of Corollary to Theorems 1-3 on page 506, and the statement
of Theorem 4, namely Weierstrass M -test, to answer some of the questions
below.

Homework 8
Due Friday, April 13
Note : There may be a quiz on Friday. Also since I am busy I am having
to give some problems from text book, please dont copy them from manual.
1. Page 495, #21 (a)
Note : Please take a look at parts (b) and (c), I recommen

Homework 1
Due Friday, March 30
Note : Problems 2 and 3 are very important. Even after you are done with
them, please keep them in your mind : they will keep being useful in exam
problems and theorems that we will be encountering. I am not giving more
pro

Homework 5
Due Friday, April 6
1. Page 493, #10. Please compare with the statement of Theorem 10,
and please dont copy from the manual.
2. Construct a bijection from N to N N.
Hint : Be warned that there could be typos etc. in this hint. It is
supposed to