Advanced Calculus
This is a second course on calculus. It will take you through the property of real numbers, as a totally ordered complete field and also as the metric completion of the rational numbers. The Bolzano Weierstrass property and that every Cauchy sequence in the real numbers is convergent are just the manifestation of the complete metric space like the real numbers or R^n . The cauchy sequence technique is extended to sequence of functions, first in pointwise convergence, then in uniform convergence. Though metric property is not explicitly mentioned, the technique from metric space is used as it is most efficient and conceptually clearer. Keeping to the aim of minimum introduction of terms, the exposition takes care of the detail in implementing the ideas used and very often intricate use of inequalities.
Materials Covered:
Real Numbers - A complete totally ordered field - Axioms for R. Inf , Sup, lim inf and lim sup. Quick review of there is only one complete totally ordered field upto isomorphism.
The complex numbers - is a complete field but not totally ordered.
Sequences. Let P be the set of positive integers. A function from P into R is called a real sequence and a function from P into the complex numbers C is called a complex sequence. In general a function from P into a set X is called a sequence in X. If a: P → X is a sequence in X then a(n) is called the nth term of the sequence and is usually written an . We may also write (a1 , a2 , ...) or (an) for the sequence. Definition of convergence of a real sequence. A complex sequence converges if and only if both its real and complex parts converge. If X is Rn , then similar definition for convergence in Rn. We can of course phrase this in terms of the distance function on Rn. We define P-1 = {1/n: n an integer in P}. Restating convergence of real sequence in terms of continuity. Proposition: Define f : {0}∪P-1 → R by f(1/n) = an and f(0) = a. Then (an) converges iff f is continuous at 0. Propertices of convergence. Convergence of sums, products and quotients follows easily from continuity. Similar proof for complex sequence. Comparison test. Examples: an tends to 0 when |a| < 1. Theorem: if (an ) converges, then (an) is bounded. (But the converse is false) Squeeze Theorem for sequences.
Monotone sequences. Prop: If (an) is a real bounded monotonic sequence, then (an) is convergent. (This requires the completeness property of R. We may also use the equivalent axiom: R is complete iff every bounded monotonic sequence has a limit). Cauchy sequence. Every Cauchy sequence is bounded. Theorem. Cauchy Principle of convergence. (an) is convergent iff (an) is Cauchy. Bolzano - Weierstrass Theorem. Every bounded sequence in Rn (hence in C) has a convergent subsequence. Continuous real valued functions on closed and bounded sets of C are bounded and attain their bounds. (This may be generalized to functions on Rn and use Bolzano Weierstrass Theorem for subsets of Rn)
Series. Definition. Partial sums. Examples. Convergence. Properties. (even for complex series.) Cauchy series. Theorem. Cauchy principle of convergence. ∑ an convergces iff ∑an is Cauchy. Proposition. If ∑an converges, then an→ 0. Example of if (an ) does not converge to 0, then the series ∑ an diverges. Tests for Convergence: 1. Proposition If ∑ an is a series of real non-negative terms, then ∑ an converges iff ( sn ) is bounded. 2. Comparison test: Let ∑ an and ∑bn be two series of real non-negative terms such that an ≤ λ bn for some real number λ. Then ∑ bn converges implies that ∑ an converges and ∑ an diverges implies that ∑ bn diverges. Conditional convergence. Alternating Series test. Ratio test (Very important test.) Integral test. The Cauchy root test.
Power series functions. Definition (to include complex series functions). Examples: exp(x), cos(x), sin(x). Domain of power series function. Continuity, differentiability and integrability of power series functions. Main Theorem. Let ∑ an xn be a power series. There are 3 alternatives. 1. ∑ an xn converges only when x = 0 and diverges everywhere; 2. ∑ an xn converges for all x and 3. There exists r such that ∑ an xn converges absolutely if |x| <r, diverges if |x| > r and may do either when |x| = r. This number r is called the radius of convergence. This number r can almost always be found using the D'Alambert's Ratio Test. (Alternatively by Cauchy's nth root test) Examples. Let D(r) be the (open) disc of convergence. Then the power series defines a continuous function f : D(r) → R. Remark. It may happen that the power series is convergent at some point on the boundary of the interval of convergence, then f is also continuous there at this point (Abel's Theorem). Differentiating power series. The power series ∑ an xn , ∑ nan xn-1, ∑ n(n-1)an xn-2 , ... All have the same radius of convergence. Theorem. Within the disk of convergence, the power series function can be differentiated term by term. Counter example to writing (infinite) Taylor series expansion for a function f. (f(x) = 1/exp(1/x2) for x ≠ 0, f(0) = 0. Using power series to define exp for the real case. Power series definition of sine, cosine. Proof of the addition formulae. Relation with the trigonometric ratios and as circular function and the definition of π.
Uniform Convergence of sequences of functions. Abstractly the proof of the above result uses uniform convergence. Sequence of functions. Pointwise convergence and uniform convergence of a sequence of functions. Cauchy Criterion (A sequence of functions converges uniformly if and only if it is uniformly Cauchy--- Cauchy in the sup norm or metric). Theorem: If fn converges uniformly to f and each fn is continuous on E, then f is continuous on E. Commutativity of the integral and limit and differentiaition and limit for uniform limit. (Proof later) Weierstrass M-test. Abel's and Dirichlet's tests for uniform convergence.
Continuous functions and limits. The extreme Value Theorem, Intermediate Value Theorem. Uniform continuity and integrability. Uniform limit of a sequence of integrable functions is integrable. Riemann’s integrability criterion. The FTC. Convergence of Darboux sums and Riemann sums. Commutativity of the integral and limit and differentiaition and limit for uniform limit (Proof). Arzela's Dominated convergence theorem for the Riemann integral. The Newton-Gregory Formula
Approximation by Taylor polynomials. The Lagrange Remainder Theorem. The convergence of Taylor polynomials. The Cauchy Integral Remainder Formula and the Binomial Expansion. The Weierstrass Approximation Theorem. Bernstein’s proof.
Lectures 4 to 10 have been replaced by typed draft chapters, comments welcomed
The lecture notes are hand written notes in pdf format
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Lecture Notes |
Tutorial |
| Lecture 1
Ref: Chapter 2 Dedekind's cuts |
Tutorial 1 Soln1 |
| Lecture 2
Lecture 2. Sequences, Cauchy's Principle of Convergence and the
Completeness of R.
|
Tutorial 2 Soln2 |
| Lecture 3 Series (Emphasis on Real Series) Part 1 |
Tutorial 3 Soln3 |
| Lecture 3 Series Part 2 |
Tutorial 4 Soln4 |
| Lecture 4 Power Series |
Tutorial 5 Soln5 |
| Lecture 5 Differentiation and power series. Consequence of uniform convergence. |
Tutorial 6 Soln6 |
| Lecture 6 Integartion, Uniform Convergence and Power Series |
Tutorial 7 Soln7 |
| Lecture 7 Weierstrass Approximation Theorem |
Tutorial 8 Soln8 |
| Lecture 8 The elementary functions: Exponential, logarithmic, sine and cosine functions |
Tutorial 9 Soln9 |
| Lecture 9 Product,Quotient and inversion of power series, analytic function |
Tutorial 10 Soln10 |
| Lecture 10. Special Tests for Convergence of Series. | |
| Lecture 11. Improper Integrals. Definition. Convergence and tests for convergence |
| Learning Guide | |
| Week 1 | From natural numbers to the real
numbers. Motivation for a definition of real numbers, from a logical
point of view and from a desirable point of view in terms of its
properties. Counting numbers to integers - the need for zero From integers to the rational numbers, its algeraic structures as a totally ordered field. The notion of positive cone, ordering on the rational numbers. The meaning of irrational numbers. Dedekind cut as a possible definition. (Detail knowledge not required.) Real numbers as a complete totally ordered field. Definition of upper and lower bounds, supremum and infimum. The completeness property. The archimedean property. The rational number field is not complete. Density of the rationals and also of the irrationals.
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| Week 2 | Sequences. Definition. Convergence of
sequence. Sequences other than real sequences, complex sequence, sequence
in Rn, etc. Equivalent definition of convergence for real sequence in
terms of continuity. Convergence of sums, products, quotient.
Comparison test, Squeeze Theorem for sequences. Monotone sequences. Important result: any bounded monotone sequence is convergent. Cauchy sequence. Theorem: A sequence in R is convergent if and only if it is Cauchy. (Cauchy principle of convergence).
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| Week 3 | Proof of Cauchy principle of convergence. The Bolzano-Weierstrass Theorem. Important Theorem. Equivalence of Completeness and Cauchy Principle of convergence and the Conclusion of Bolzano Weierstrass Theorem. Subsequences. Technical result: any sequence has a monotone subsequence. Proof of the Bolzano-Weierstrass Theorem.
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| Week 4 | Series. Definition of n-th partial sums
and convergence of a series. New meaning of the "=" sign to denote
the limit of the series. Example: Geometric series. Notation
and use of the summation sign. Definition of a cauchy series. A series is convergent if and only if it is a cauchy series. (Cauchy principle of convergence). If ∑an converges, then an → 0. Thus if an does not converge to 0, then ∑an diverges. If ∑an is a series of non-negative terms, then it is convergent if and only if it is bounded. Comparison test for series with non-negative terms. Example: ∑ 1/n2 is convergent. If ∑ |an| is convergent, then ∑an is convergent. Definition of absolute convergence. Examples. ∑ (-1)n+1 / n is convergent but not absolutely convergent. Definition of conditional convergence. Leibnitz's Alternating series test. A very important test D' Alembert Ratio Test (for absolute convergence). A more refined version. Examples of absolute convergence and conditional convergence. The Integral Test. The convergence and divergence for ∑ 1/np for p > 0. Example of the use of comparison test . The Cauchy Root Test and a refined version using lim sup. Example. ∑(-1)n+1/n = ln(2).
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| Week 5 | Power Series. Definition and examples.
Domain of convergence, radius of convergence and the disk of convergence.
Proof of the basic theorem for series: it either converges only at 0, for
all x or converges absolutely inside a disk of radius r and diverges
outside a disk of radius r and may do either on the boundary circle of
convergence. Logarithmic series. The standard test for convergence is
still the ratio test. Examples of the use of this test for various power
series. Continuity of power series functions. It is continuous inside the (open) disk of convergence. For real power series, continuity at the end point of the interval of convergence is assured by Abel's Theorem if it is convergent there. Proof much later
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| Week 6 | Sequence of functions. Pointwise
convergence and uniform convergence of a sequence of functions.
Examples of pointwise convergence. Theorem If fn
converges uniformly to a function f on E, and if each fn
is continuous, then f is also continuous. Another proof of the continuity
of power series function using uniform convergence. Using the above
theorem to deduce non-uniform convergence. Examples. Questions arising from the above theorem. Commutation of two different kinds of limiting process. Determining radius of convergence. Using the standard Ratio Test . Using the Cauchy-Hadamard formula.
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| Week 7 | Weierstrass M-Test for uniform convergence of
a series of functions. Definition of a uniform Cauchy sequence of
functions. A sequence of (real valued) functions converges
uniformly if and only if it is uniformly Cauchy (Cauchy principle of
convergence) [ A sequence of functions in the space of bounded real valued
functions is convergent if and only if it is a Cauchy sequence with
respect to the sup metric. Ref: Binmore, Foundation of analysis Book 2
Topological Ideas.] Example of the use of the Weierstrass M-Test.
Example of a sequence of differentiable function fnconverging
uniformly to a function f but the derivative fn' does
not converge to f '. We need
additional requirement on fn' to deduce that
fn' converges to f '. For this we can bring
the result about integrating a sequence of continuous function. If gn is a sequence of continuous function converging uniformly to g on [a, b], then the sequence of integrals,∫gn from a to b converges to ∫ g from a to b. With this theorem we can then prove the following: If fnconverges pointwise to a function f and fn' converges uniformly to a function g, then f ' = g. Example: the exponential series. Application to power series function. The two power series obtained from a fixed one by differentiating or integrating its terms, term by term have the same radius of convergence as the given one. Consequence: we can differentiate a power series infinitely many times within its interval convergence. Example of the use of Taylor's Theorem to find a solution to the differential equation f ' = f and f(0) =1. Example of a function that does not have a power series expansion. Condition of convergence of Taylor polynomials to give an infinite power series expansion fo a given function. A sufficient condition for Taylor series expansion. Example: the sine series expansion. Application: Proof that e is irrational. Abels' Theorem and continuity of power series function at the boundary point of the disk of convergence. |
| Week 8 | Integration and uniform convergence.
Review of Riemann integration theory. Darboux sums, upper integral
and lower integrals. Characterization of Riemann integrability. Theorem:
the uniform limit of a sequence of Riemann integrable functions is
integrable. In particular the integral of the limiting function is
the limit of the sequence of integral of the term of the sequence.
Abel's test for the uniform convergence of infinite sums of series of
functions. Example. Dirichlet's test for the uniform
convergence of infinite sums of series of functions. Example: Simple
trigonometric series ∑ansin(nx) , Power series expansion
for integrals with no closed form. Application: expansion for π/4 .
A slight generalization of convergence theorem for sequence of integrals of functions, which form a sequence that converges only pointwise: The Arzela's Dominated Convergence Theorem. Application and examples. Consequence of uniform convergence. Convergence of ∑ sin(nx)/ns, ∑ ansin(nx), ∑ancos(nx). Newton's Binomial series. |
| Week 9 | Weierstrass Approximation Theorem.
Proof via Bernstein polynomial technique. The elementary functions: exponential, logarithmic, sine and cosine functions defined analytically. Proof that they are the same as the usually defined functions. Properties and uniqueness of the sine and cosine functions. Definition of π (Richard Baltzer, Landau). Sine and cosine thus defined and the trigonometric ratios. |
| Week 10 | (Optional) Product of power series. Term wise product versus cauchy product. Multiplication, quotient and inversion of Power series. Special Tests for convergence of series. Kummer's Test, Raabe's Test, Gauss Test, Rabbe's type test and other more refined test. Improper integrals. Definitions and examples. (For the first time Lebesgue integral is introduced) Improper Riemann integrals. Convergence Criteria and relation with Lebesgue integrals. Absolute and conditional convergence of improper Riemann integrals. Test for convergence. Examples. |