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MA10229: Analysis 1A

Academic Year: 2018/9
Owning Department/School: Department of Mathematical Sciences
Credits: 6      [equivalent to 12 CATS credits]
Notional Study Hours: 120
Level: Certificate (FHEQ level 4)
Period:
Semester 1
Assessment Summary: EX 100%
Assessment Detail:
  • Examination (EX 100%)
Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
Requisites: While taking this module you must take MA10209
In taking this module you cannot take MA10207 . You must have grade A in A-level Mathematics or equivalent in order to take this unit.
Description: Aims:
To define the notions of convergence and limit precisely and to give rigorous proofs of the principal theorems on the analysis of real sequences.

Learning Outcomes:
After taking this unit, the student should be able to:
* State definitions and theorems in parts of real analysis;
* Present proofs of the main theorems;
* Apply these definitions and theorems to simple examples;
* Construct their own proofs of simple unseen results.

Skills:
Numeracy T/F A, Problem Solving T/F A, Written and Spoken Communication F (in tutorials).

Content:
Quantifiers. Definitions; sequence, limit. Numbers, order, absolute value, triangle inequality, binomial inequality. Convergence, divergence, infinite limits. Examples: 1/n, an. Algebra of limits. Uniqueness of limits. Growth factor. Convergent sequences are bounded. Axiom: bounded monotone sequences converge. Sequence converging monotonically to root 2. Observations: roots generally not algebraically constructible, transcendental functions are defined as limits. Subsequences, Bolzano-Weierstrass Theorem. Cauchy sequences.
Convergence of series. Geometric series. Comparison and Ratio tests. Harmonic series; condensation. Absolute and conditional convergence. Leibniz's Theorem (alternating series).
Nested intervals. Application: uncountability of R. Countability of Q. Sup and inf via convergence of bounded monotonic sequences. Limsup and liminf. Existence of n-th roots, definition of rational powers. Infinite decimals.
Programme availability:

MA10229 is only available subject to the approval of the Director of Studies.


Notes:

  • This unit catalogue is applicable for the 2018/19 academic year only. Students continuing their studies into 2019/20 and beyond should not assume that this unit will be available in future years in the format displayed here for 2018/19.
  • Programmes and units are subject to change in accordance with normal University procedures.
  • Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules.
  • Undergraduates: .
  • Postgraduates: .