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 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