


Calculus
Contents B 





Sequences
and limits

Infinite
sequences 
Sequences notation 
the
rule for the nth
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
nth
term test for divergence 
Properties
of series 
The
product of two series or the Cauchy product 
Geometric
series 
Pseries 
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 lefthand limit) and a limit
on the right (a righthand 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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