Calculus - I
Calculus Contents B
      Sequences and limits
      Infinite sequences
         Sequences notation - the rule for the n-th term of a sequence
         Graphing the terms of a sequence on the number line
      The limit of a sequence
         The definition of the limit of a sequence
         Convergence of a sequence
         Verifying the convergence of a sequence from the definition examples
      Limits of sequences
      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)
      The definition of the real number e
     The limit of a sequence theorems
      The cluster point or accumulation point
         Divergent sequences
      Sufficient condition for convergence of a sequence
         The Cauchy criterion (general principle of convergence)
      Some important limits
      Operations with limits
         Operations with limits examples
      Series
      Infinite series
         The sequence of partial sums
         The sum of the 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
         A limit on the left (a left-hand limit) and a limit on the right (a right-hand limit)
      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)
         Squeeze rule
         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
         Use of the composition rule to evaluate limits of functions
 
 
 
 
 
 
 
 
 
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