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| Calculus
Contents B |
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Sequences
and limits
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Infinite
sequences |
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Sequences notation -
the
rule for the n-th
term of a sequence |
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Graphing the terms of a sequence on the number line |
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The limit of a sequence |
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The definition of the limit of a sequence |
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Convergence of a sequence |
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Verifying the convergence of a sequence from the
definition examples |
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Limits of sequences |
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Properties of
convergent sequences |
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The limit value is exclusively determined by the behavior of the terms
in its close neighborhood |
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Bounded
sequences |
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Every convergent sequence is bounded |
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Increasing,
decreasing, monotonic sequence |
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Every subsequence of
a convergent sequence
converges to the same limit |
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Every
bounded monotonic sequence is convergent example |
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Sandwich
theorem (result) or squeeze rule |
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Least
upper bound (or supremum, abbrev. lub, sup) and greatest lower bound
(infimum,
or glb, inf) |
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The
definition of the real number e |
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The limit of a sequence theorems |
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The
cluster point or accumulation point |
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Divergent
sequences |
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Sufficient condition for convergence of a sequence |
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The Cauchy criterion (general principle of convergence) |
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Some
important limits |
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Operations
with limits |
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Operations
with limits examples |
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Series
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Infinite series |
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The sequence of partial
sums |
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The
sum of the series |
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Convergence
of infinite series |
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Divergence
of infinite series |
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Convergent
and divergent series
examples |
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Harmonic
series |
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The remainder
or tail
of the series |
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Necessary
and sufficient
condition for the convergence of a series |
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Necessary
condition for the convergence of a series |
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The
n-th
term test for divergence |
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Properties
of series |
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The
product of two series or the Cauchy product |
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Geometric
series |
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P-series |
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Alternating
series |
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Alternating series test
or Leibnitz's
alternating series test |
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Absolute
convergence |
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Conditional
convergence |
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Series
of positive terms |
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Tests
for convergence |
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Comparison
test |
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Limit
comparison test |
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Ratio
test |
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Root
test or Cauchy's root test |
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Power
series |
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Power
series or polynomial with infinitely many terms |
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Maclaurin and Taylor series |
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The power series expansion of the exponential
function |
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Properties
of the power series expansion of the exponential function |
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The radius of convergence or the interval of convergence |
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The power series expansion of the logarithmic function |
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Properties
of the power series expansion of the logarithmic function |
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The power series expansion of the sine function |
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Properties
of the power series expansion of the sine function |
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The power series expansion of the cosine function |
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Properties
of the power series expansion of the cosine function |
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The power series expansion of the hyperbolic sine and hyperbolic cosine function |
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Properties
of the power series expansion of the hyperbolic sine and hyperbolic cosine function |
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The
binomial series
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The binomial
series expansions to the power series
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The binomial
series expansion to the power series example
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The
limit of a function |
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The
definition of the limit of a function |
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A limit
on the left (a left-hand limit) and a limit
on the right (a right-hand limit) |
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Continuous
function |
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Limits
at infinity (or limits of functions as x approaches
positive or negative infinity) |
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Infinite limits |
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The
limit of a function examples |
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Vertical, horizontal
and slant (or oblique) asymptotes |
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Monotone
functions - increasing or decreasing in value |
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Limit of a function
properties (theorems or laws) |
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Squeeze
rule
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Composition
rule
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Limits of
functions
properties use, examples |
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Limits of
rational functions |
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Evaluating
the limit of a rational function at infinity |
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Evaluating
the limit of a rational function at a point |
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The limit of a rational function that is defined at the given point |
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The limit of a rational function that is not defined at the
given point |
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The limit of a rational function at infinity
containing roots (irrational expressions) |
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The limit of a rational function at
a point
containing irrational expressions, use of substitution |
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Evaluating
the limit of a rational function
containing irrational expressions using rationalization |
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Limits of
trigonometric functions |
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Evaluating
trigonometric
limits examples |
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Limits of functions
based on the definition of the natural number e |
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Evaluating
limits of functions
based on the definition of the natural number e |
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Use
of the composition rule to evaluate limits of functions |
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