Calculus - I
Calculus Contents B
Sequences and limits Infinite sequences The limit of a sequence Properties of convergent sequences
The limit value is exclusively determined by the behavior of the terms in its close neighborhood
Bounded sequences
Every convergent sequence is bounded
Increasing, decreasing, monotonic sequence
Every subsequence of a convergent sequence converges to the same limit
Every bounded monotonic sequence is convergent example
Sandwich theorem (result) or squeeze rule
Least upper bound (or supremum, abbrev. lub, sup) and greatest lower bound (infimum, or glb, inf) e The cluster point or accumulation point
Divergent sequences Sufficient condition for convergence of a sequence Some important limits Operations with limits
Operations with limits examples
Series Infinite series
Convergence of infinite series
Divergence of infinite series
Convergent and divergent series examples Harmonic series The remainder or tail of the series Necessary and sufficient condition for the convergence of a series
Necessary condition for the convergence of a series
The n-th term test for divergence Properties of series
The product of two series or the Cauchy product Geometric series P-series Alternating series
Alternating series test or Leibnitz's alternating series test
Absolute convergence
Conditional convergence Series of positive terms Tests for convergence
Comparison test
Limit comparison test
Ratio test
Root test or Cauchy's root test
Power series Power series or polynomial with infinitely many terms
Maclaurin and Taylor series The power series expansion of the exponential function
Properties of the power series expansion of the exponential function
The radius of convergence or the interval of convergence The power series expansion of the logarithmic function
Properties of the power series expansion of the logarithmic function The power series expansion of the sine function
Properties of the power series expansion of the sine function The power series expansion of the cosine function
Properties of the power series expansion of the cosine function The power series expansion of the hyperbolic sine and hyperbolic cosine function
Properties of the power series expansion of the hyperbolic sine and hyperbolic cosine function The binomial series
The binomial series expansions to the power series
The binomial series expansion to the power series example
The limit of a function The definition of the limit of a function Continuous function
Limits at infinity (or limits of functions as x approaches positive or negative infinity)
Infinite limits
The limit of a function examples Vertical, horizontal and slant (or oblique) asymptotes
Monotone functions - increasing or decreasing in value Limit of a function properties (theorems or laws)

Composition rule
Limits of functions properties use, examples Limits of rational functions
Evaluating the limit of a rational function at infinity
Evaluating the limit of a rational function at a point
The limit of a rational function that is defined at the given point
The limit of a rational function that is not defined at the given point
The limit of a rational function at infinity containing roots (irrational expressions)
The limit of a rational function at a point containing irrational expressions, use of substitution
Evaluating the limit of a rational function containing irrational expressions using rationalization Limits of trigonometric functions
Evaluating trigonometric limits examples Limits of functions based on the definition of the natural number e
Evaluating limits of functions based on the definition of the natural number e    