Course Page Analysis 1 (2WA30)
"The
noblest pleasure is the joy
of understanding."
(L.
da Vinci)
Lecturer in charge: G. Prokert
Instructors: M. Anthonissen, J. Portegies, R. Kaasschieter / G
Prokert, J. ten Thije Boonkkamp
General remarks
A main objective of the course is to give an introduction not only to the
subject matter but also to the typically
mathematical way of thinking and working. In particular, we will
focus on how to prove propositions and on reading, writing and
understanding the "language of mathematics".
This is a challenge for everybody who starts a serious study of mathematics,
as this is often a new approach, in comparison to highschool mathematics. It
is not exceptional but absolutely common that difficulties
arise at this point.
Some practical tips to cope with these difficulties:
- Practice makes perfect. Nobody expects that he/she will learn
to skate by just watching a skater, or to play the piano by listening to
a pianist. The same holds for learning mathematics.
Doing (yourself! at least!) the exercises accompanying the lectures is
the central part of the course.
- Keep calm and carry on. As with skating or playing the piano,
you will usually not immediately be on the level of an expert. This is
just normal. If you keep working, you will soon experience progress.
- Keep up with the lectures. It is typical for our course that
later lectures depend and rely on the contents of the earlier ones.
Avoid falling behind the lectures in your own work on understanding and
getting acquainted with the contents. Waiting for problems to vanish by
themselves is no feasible strategy.
- Formulate questions. When you encounter unclear points, try to
formulate a question about it - as precisely as possible. Often this is
a significant step in resolving the issue.
- Talk about it. You are not alone. One of the aims of having you
work in groups on the exercises is to have you talk to each other about
mathematics. Do this, also to your other fellow students, instructors,
...
- Look around. For the contents of our lectures, books
as well as material on the internet is abundantly available. Try to
consult such sources as well, and try to learn from
the similarities as well as from the differences.
For a start, using these
best practices will be helpful.
Examination
- Final test (written, 70 %)
- Intermediate test (written, 10%): Wednesday December 4,
8:45-10:15
- Homework (to be handed in weekly in groups, 20%)
Contents per lecture
(The precise distribution of the material over the lectures may be subject
to change!)
[K] denotes the course textbook: W. Kosmala, A friendly introduction to
Analysis, 2nd edition, Prentice Hall, ISBN: 0131273167
Lecture 1: Ordered sets, ordered
fields, real numbers I
Material: [K] 1.7.
Core concepts: order
relation, ordered set, upper / lower bound, maximum, minimum, infimum,
supremum, open / half-open / closed interval, ordered field, complete
ordered field
Core theorems: The rational numbers
form an ordered, but not completely
ordered field. The real numbers form a completely ordered field.
Abilities: Finding maxima / minima and suprema / infima, in
particular for sets of real numbers, working with these concepts and proving
corresponding properties.
Homework Lecture 1
Lecture 2: Real numbers II,
sequences I
Material: [K] 1.7, 2.1
Core concepts: sequence, bounded sequence (from above /
below), (strictly) increasing / decreasing sequence, limit
Core theorems: Uniqueness of the limit, boundedness of convergent
sequences
Abilities: recognizing properties of sequences, working with the
formal limit definition
Homework Lecture 2
Lecture 3: Sequences II: Limit
theorems and standard limits
Material: [K] 2.2 - 2.4
Core concepts: improper limits
Core theorems: Limit theorems for
sums, products, quotients, inequalities, squeeze theorem, theorems on
standard limits
Abilities: Finding limits on the basis of the theorems discussed
Homework Lecture 3
See also the Lightboard videos on CANVAS! (access via menu item
"modules")
Lecture 4: Sequences III:
Subsequences and Cauchy sequences
Material: [K] 2.5 - 2.6
Core concepts: Number e,
subsequence, accumulation point of a sequence (called "subsequential limit
point" in [K]. Mind the difference with the definition [K] 2.5.2. of
"accumulation point" for a set of points used in [K], see p.108), Cauchy
sequence
Core theorems: Bolzano-Weierstrass theorem, Cauchy sequences are
convergent (in R)
Abilities: applying the
discussed theorems in convergence proofs
Homework Lecture 4
Lecture 5: Series I
Material: [K] 7.1, 7.2 (in part)
Core concepts: partial sum of a sequence, series, convergence /
divergence of a series, geometric series, (hyper)harmonic series, majorant,
minorant, absolute convergence
Core theorems: Convergence behavior of geometric and
(hyper)harmonic series, absolutely convergent series are convergent, the
terms of a convergent series approach zero, the "tail" of a convergent
sequence approaches zero.
Abilities: recognizing series of the types discussed here, applying
the discussed theorems in the investigation of convergence of series
Homework Lecture 5
Lecture 6: Series II
Material: [K] 7.3, 7.4
Core concepts: alternating series
Core theorems: ratio test, root test, Leibniz criterion
Abilities: recognizing series of the types discussed, applying the
discussed theorems in the investigation of the convergence of series
Homework Lecture 6 (For the
references to [K], see CANVAS)
Lecture 7: Series III
Material: notes, [K] 7.4
Core concepts: Cauchy
product, exponential function, reordering of a series
Core theorems: Product of two absolutely convergent series,
properties of the exponential function, theorems om reordering of absolute /
not absolute convergent series
Abilities: calculations with the exponential function. applying the
discussed theorems
Homework Lecture 7 (not to be handed
in)
Lecture 8: Limits of functions, continuity (revisited)
Material: [K] Ch.
3, 4
Core
concepts:
Limit of a function at a point and at +/- infinity, improper limits,
continuity
Core theorems: Limit theorems for
functions, continuity in terms of sequences, sums/products/compositions of
continuous functions are continuous, maximum property, intermediate value
theorem
Abilities: Finding limits, proving
of corresponding properties, recognizing / proving continuity of functions,
applying the discussed theorems
Homework Lecture 8
Lecture 9: Differentiable functions,
Taylor's theorem (revisited)
Material:
[K] Ch. 5
Core concepts: differentiability, derivative, linearization, higher
order derivatives
Core theorems: Rolle's theorem,
mean value theorem, necessary local optimality condition, Taylor's theorem
Abilities: applying these theorems,
for example in determining limits, extrema etc.
(For the reference to [K], see CANVAS)
Lecture 10: Sequences of functions I
Material: [K] 8.1,8.2, notes
Core concepts: sequence of
functions, pointwise / uniform convergence
Core theorems: uniform convergence preserves limits and continuity
Abilities: investigating
pointwise and uniform continuity of sequences of functions
Homework Lecture 10
(For the references to [K], see CANVAS)
Lecture 11: Sequences of functions
II, function series I
Material: [K] 8.2,8.3,8.4, notes, Sheets
Core concepts: function series
Core theorems: Uniform convergence of the derivatives preserves
differentiability, Weierstrass' en Dini's criteria for uniform convergence
Abilities: investigating convergence
of function series, applying the discussed theorems
(For the references to [K], zie CANVAS)
Lecture 12: Function series II, power series
Material: [K] 8.4,8.5
Core concepts: power series
Core theorems: Criteria for
continuity and differentiability of function series
Abilities: recognizing
continuity and differentiability of function series, investigating
convergence of power series
(For the references to [K], zie CANVAS)
Lecture 13: Convergence of power series
Material: [K] 8.5, 8.6, notes
Core concepts: limes superior,
convergence radius of a power series
Core theorems: Characterization(s) of the convergence radius,
corresponding convergence behavior of a power series
Abilities: Finding the radius
of convergence of a power series
(For the references to [K], zie CANVAS)
Lecture 14: Calculations with power series
Material: [K] 8.5, 8.6
Core theorems: power series converge
uniformly on each bounded and closed interval within the open interval of
convergence,
power series represent infinitely differentiable functions within their
convergence interval, identity theorem for power series, power series are
identical with their Taylor series
Abilities: applying the
rules for calculation with power series, "comparison of coefficients" for
solving linear differential equations
Homework Lecture 14
(not to be handed in)
List of standard series
Quiz: True or false?
Sheets from the lectures: