Higher Mathematics for Physics and Engineering
1 Preliminaries
1.1 Basic Notions of a Set.
1.1.1 Set and Element
1.1.2 Number Sets….
1.1.3 Bounds
1.1.4 Interval
1.1.5 Neighborhood and Contact Point…..
1.1.6 Closed and Open Sets..
1.2 Conditional Statements
1.3 Order of Magnitude…..
1.3.1 Symbols O, o, and~.
1.3.2 Asymptotic Behavior
1.4 Values of Indeterminate Forms
1.4.1 l’Hôpital’s Rule…
1.4.2 Several Examples.
2Real Sequences and Series
2.1 Sequences of Real Numbers
2.1.1 Convergence of a Sequence
2.1.2 Bounded Sequences.. 2.1.3 Monotonic Sequences
2.1.4 Limit Superior and Limit Inferior.
2.2 Cauchy Criterion for Real Sequences
2.2.1 Cauchy Sequence. 2.2.2 Cauchy Criterion
2.3 Infinite Series of Real Numbers.
2.3.1 Limits of Infinite Series
2.3.2 Cauchy Criterion for Infinite Series X Contents 2.3.3 Absolute and Conditional Convergence. 2.3.4 Rearrangements..
2.4 Convergence Tests for Infinite Real Series.
2.4.1 Limit Tests
2.4.2 Ratio Tests.
2.4.3 Root Tests
2.4.4 Alternating Series Test.
3 Real Functions
3.1 Fundamental Properties
3.1.1 Limit of a Function. 3.1.2 Continuity of a Function
3.1.3 Derivative of a Function.
3.1.4 Smooth Functions
3.2 Sequences of Real Functions
3.2.1 Pointwise Convergence
3.2.2 Uniform Convergence
3.2.3 Cauchy Criterion for Series of Functions.
3.2.4 Continuity of the Limit Function
3.2.5 Integrability of the Limit Function.
3.2.6 Differentiability of the Limit Function.
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