USMA-AKB06 : BSc (hons) Statistics (Full-time with Thick Sandwich Placement) - Year 1.USMA-AFB05 : BSc (hons) Statistics (Full-time) - Year 1.USMA-AAB14 : BSc (hons) Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 1.USMA-AAB02 : BSc (hons) Mathematics and Statistics with Study Year Abroad (Full-time with Study Year Abroad) - Year 1.USMA-AKB02 : BSc (hons) Mathematics and Statistics (Full-time with Thick Sandwich Placement) - Year 1.USMA-AFB01 : BSc (hons) Mathematics and Statistics (Full-time) - Year 1.USMA-AKB14 : BSc (hons) Mathematics (Full-time with Thick Sandwich Placement) - Year 1.
USMA-AFB13 : BSc (hons) Mathematics (Full-time) - Year 1.USMA-AAB16 : BSc (hons) Mathematical Sciences with Study Year Abroad (Full-time with Study Year Abroad) - Year 1.USMA-AKB16 : BSc (hons) Mathematical Sciences (Full-time with Thick Sandwich Placement) - Year 1.USMA-AFB15 : BSc (hons) Mathematical Sciences (Full-time) - Year 1.USCM-AAM14 : MComp (hons) Computer Science and Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 1.USCM-AKM14 : MComp (hons) Computer Science and Mathematics with Industrial Placement (Full-time with Thick Sandwich Placement) - Year 1.USCM-AFM14 : MComp (hons) Computer Science and Mathematics (Full-time) - Year 1.USCM-AAB20 : BSc (hons) Computer Science and Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 1.USCM-AKB20 : BSc (hons) Computer Science and Mathematics (Full-time with Thick Sandwich Placement) - Year 1.USCM-AFB20 : BSc (hons) Computer Science and Mathematics (Full-time) - Year 1.MA10207 is Compulsory on the following programmes: Department of Computer Science
#Sandwich sequences convergence series#
Binomial Theorem for real exponent by differentiating quotient of series by (1+x) p.
Logarithmic series on (-1,1) from geometric series. Sketch extension to complex arguments and exp(iz)=cos(z)+isin(z). Uniqueness for Cauchy problem for y'' = -y by differentiating (y')²+y², trigonometric addition formulae. Series definitions of exponential and trigonometric functions. Uniform Convergence uniform limit of continuous functions uniform Cauchy sequences Weierstrass M-test. Sign of second derivative maxima and minima, convexity. Taylor's Theorem with Lagrange remainder. Continuous inverse of strictly increasing (continuous) function on interval. Algebra of limits, continuous functions, polynomials. Infinite decimals.Ĭontinuity and limits of functions of real variables. Existence of n-th roots, definition of rational powers. Sup and inf via convergence of bounded monotonic sequences. Subsequences, Bolzano-Weierstrass Theorem. Observations: roots generally not algebraically constructible, transcendental functions are defined as limits. Sequence converging monotonically to root 2. Axiom: bounded monotone sequences converge. Growth factor.Ĭonvergent sequences are bounded. Numbers, order, absolute value, triangle inequality, binomial inequality.Ĭonvergence, divergence, infinite limits. Written and Spoken Communication F (in tutorials and problem sheets). * Construct their own proofs of simple unseen results.
* Apply these definitions and theorems to simple examples. * State definitions and theorems in real analysis. To define the notions of convergence, limit and continuity precisely and to give rigorous proofs of the principal theorems on the analysis of real sequences and real functions of a real variable.Īfter taking this unit, the students should be able to: Students must have a grade A in A-level Mathematics or equivalent in order to take this unit. Requisites: While taking this unit you must take MA10209 and take MA10210. Supplementary Assessment: MA10207 Mandatory extra work (where allowed by programme regulations) Owning Department/School: Department of Mathematical Sciences This is a result of Bernoulli's inequality, which stipulates that for all n in the natural numbers, (1 + x)^n >= 1 + nx, and that x >= -1.MA10207: Analysis 1 Academic Year: 2012/3 Now, let's look at the sequence b n = (1 + a/n)^n, which converges to e^a as n gets closer to infinity. The given sequence must also converge to 1 in accordance with the sandwich theorem. In particular, we shall demonstrate that the provided sequence and two other sequences, one above and one below it, both converge to 1. The sandwich theorem can be used to demonstrate that for every a > 0, the sequence of nth roots of a converges to 1.
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