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MA10229: Analysis 1A

Follow this link for further information on academic years Academic Year: 2018/9
Further information on owning departmentsOwning Department/School: Department of Mathematical Sciences
Further information on credits Credits: 6      [equivalent to 12 CATS credits]
Further information on notional study hours Notional Study Hours: 120
Further information on unit levels Level: Certificate (FHEQ level 4)
Further information on teaching periods Period:
Semester 1
Further information on unit assessment Assessment Summary: EX 100%
Further information on unit assessment Assessment Detail:
  • Examination (EX 100%)
Further information on supplementary assessment Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
Further information on requisites 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.
Further information on descriptions 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.
Further information on programme availabilityProgramme availability:

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


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