MA10229: Analysis 1A
[Page last updated: 15 October 2020]
Academic Year: | 2020/1 |
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
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Assessment Summary: | EX 100% |
Assessment Detail: | |
Supplementary Assessment: |
- Like-for-like reassessment (where allowed by programme regulations)
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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.
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Notes: - This unit catalogue is applicable for the 2020/21 academic year only. Students continuing their studies into 2021/22 and beyond should not assume that this unit will be available in future years in the format displayed here for 2020/21.
- 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.
- Find out more about these and other important University terms and conditions here.
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