Course: MAT 376 3.0 Real Analysis II (Compulsory)
Course Content: Properties of the real numbers: The real Number System, Algebraic Structure, Order Structure, Bounds, Sups and Infs, The Archimedean property, Inductive property of , The rational numbers are dense, The metric structure of ; Sequences: Real sequences, bounded sequences and monotone sequences, Convergence of sequence and their limits, Algebra of convergent sequence, Subsequences, Cauchy condition for convergence of sequences, Upper and Lower limits (limsup and liming); Series: Real series and bounded series, Convergent series and Algebra of convergent series, Cauchy condition for convergence, Series of positive terms, Rearrangement of series, Ratio test, Root test and Condensation test, Absolute convergent and conditional convergent, Rearrangement theorem for absolutely convergent series; Real Functions: Limit of a function and Algebra of limits, Continuity and Algebra of continuous of functions, Maxima, minima and Intermediate value theorem, Uniform continuity; Differentiable Functions: Differentiability, Rolle’s Teeorem and Mean value theorem, Maclaurin’s Series, L’Hospital’s rules for indeterminate forms; Sequences and Series of Functions: Power Series and Radius of convergence, Point wise convergence, Uniform convergence, Weierstrass M- Test, The Taylor series, Trigonometric series
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