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# Department of Mathematical Sciences

# MATH1051 Analysis I

This course deals mainly with "limits of infinite processes". It provides a firm foundation for the operations of differentiation and integration that you already know something about. In addition, you will learn how to answer questions such as the following:

• What is the limit of the sequence (2/1)¹, (3/2)², (4/3)³, (5/4)⁴,… of rational numbers?
• It is not hard to believe that the geometric series 1+1/2+1/4+1/8+… converges to the value 2, but what does the series 1+1/2+1/3+1/4+… converge to?
• What is the value of the integral of x²/(1+x³) over 0 to ∞?

We shall discuss techniques for answering questions of this sort. But analysis consists of more than simply problem-solving. Ultimately, it is about constructing logical arguments (proofs), using the correct language and style, and what mathematicians call rigour. Acquiring this skill is more important than learning problem-solving tricks, but also more difficult, especially at first. We hope that by the end of the year, you will be able to invent and write out simple proofs.

## Outline of Course

Aim: An understanding of the real and complex number systems, an introduction to series and limits and becoming familiar with the concept of continuity. To become familiar with sequences of functions, the concepts of differentiability, series and power series.

### Term 1

• Introduction
• Basic logic and sets: Mathematical statements and connectives (AND, OR, NOT, ...), notation and basic concepts for sets, basic operations on sets and De Morgan's rule.
• Numbers: The number systems (ℕ, ℤ, ℚ, ℝ) and the complex numbers ℂ. |x|<c ⇔ -c<x<c, |a|+|b|≥|a+b|≥||a|-|b|| for real (and complex) numbers.
• More logic: Quantifiers, negation of statements.
• Basics about sequences and limits: Notion of sequence, definition of ℯ, definition of limit and basic theorems (uniqueness of limits, COLT, pinching theorem).
• Sup and inf: ℚ, ℝ and the completeness axiom. Lim sup and Lim inf of subsets of ℝ and of real valued functions. Relation to maxima/minima. sup(f)+sup(g)≥sup(f+g) ≥sup(f)+inf(g).
• Proof techniques: Principle of indirect proof and contrapositive proof technique.
• More on limits of sequences: Accumulation points of a sequence. Notions of liminf and limsup. Bounded monotonic sequence tends to a limit. Bolzano-Weierstrass theorem (bounded sequences contain a convergent subsequence). Cauchy sequences.
• Functions, limits and continuity: Functions as maps between sets. Preimage of a set under a function. Limit of a function as x tends to infinity, the limit of xᵃ/ℯˣ as x→∞, the limit of log(x)/xᵃ as x→∞. Definition of continuity. Sum, composite of continuous functions is continuous. Intermediate value theorem and applications. Bisection proof of max-min theorem.

### Term 2

Aim: Having learnt sequences and basic properties of functions in the first term, to become familiar with sequences of functions, the concepts of differentiability, series and power series.

• Sequences of functions: Pointwise and uniform convergence, continuity of limits of uniformly convergent continuous functions.
• Differentiability: Definition. Differentiability implies continuity. Rules of differentiation. Rolle’s theorem, Mean Value theorem and applications, Newton-Raphson iteration.
• Convergence of series: Infinite series and series as sequences; convergence, examples including ∑n⁻ᵃ. Comparison test, absolute convergence theorem (absolutely convergent series are convergent), conditional convergence, rearrangements of series (Riemann series theorem), Cauchy product, ratio test, alternating sign test.
• Integration: Brief discussion of Riemann sums. Fundamental theorem of calculus. |∫f|≤∫|f|. Convergence of ∫f, comparison test, absolute convergence theorem, examples. Convergence of integrals with bounded range but unbounded integrand, comparison test, absolute convergence, examples. Formula for differentiation under the integral sign. (Integral test for convergence of series).
• Power series: Radius of convergence. Weierstrass test of uniform convergence, uniform convergence of power series. Term-by-term differentiation and integration with examples to show these results are not necessarily true for general (pointwise convergent) series of functions. Taylor series.

### Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.