real analysis flashcards
DESCRIPTION
Flashcards of mathematical theorems and formulae associated with real analysis and calculus concepts.TRANSCRIPT
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Triangle Inequality Definition of Convergence of
Sequences
Bolzano Weierstrass Theorem Cauchy Sequences
Contractive Sequences Order Hierarchy
Surjection Injection
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The sequence converges to L as if, given any > 0, there exists N such that
< >
If there is no such number the series diverges.
+ +
The sequence is called a Cauchy sequence if, given any >0, there exists N such that
< , >
A sequence converges iff it is a Cauchy sequence.
Every bounded real sequence contains a convergent subsequence.
1 log !
, > 1, > 0
A sequence is contractive if
+2 +1 < +1
where is a positive constant strictly less than 1. Such sequences converge.
If every element of B which is the image of something in the set A is the image of precisely one element of A.
If every element of B is the image of something in the set A. The range equals the codomain.
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Definition of Convergence of Functions at a point
= ?
=
= ?
=
= ?
+
= ?
Continuity Uniform Convergence of Functions
Intermediate Value Theorem Differentiability
Rolles Theorem Mean Value Theorem
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lim
1 = 1
lim0
sin
= lim
0
tan
= 1
lim0
log1
= lim
log
= 0
lim
1 +
=
We say that as if, given any > 0, there exists > 0such that
() <
whenever 0 < < .
is said to converge uniformly on an interval to a continuous function () if, given any > 0 there exists such that
() () <
whenever > and .
A function () is continuous at = if () () as .
A function is differentiable at the point = if
() = () + ( ) +
where L is independent of x and
/( ) 0 as
If is continuous on the closed finite interval [, ], then for every real number between () and (), there exists at least one point [, ] for which () = .
If is continuous on the closed interval [, ] and differentiable on the open interval (, ), then there exists at least one point (, )such that
= ()( )
If a function is defined on [, ], continuous on [, ], is differentiable on (, ), and () =(), then there exists (, )such that = 0.
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LHpitalsRule Taylor Polynomial
Lagranges Remaindersin(y)
cos(y)
sinh(y), cosh(y)and their derivatives
Other usefulHyperbolic Function Identities
arctanh(t)and its derivative
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If () has at least n derivatives at = , the th order Taylor polynomial for () about = is
=0
()( )
!
If and are differentiable in an open interval containing , () = () = 0, & () 0 in except possibly at = , then
lim
()
()= lim
()
()
if the limit on the right exists.
sin =1
2
cos =1
2 +
= +1 ()( )+1
+ 1 !
cosh2 sinh2 = 1
sech2 = 1 tanh2
tanh = 1 tanh2
sinh(2) = 2 sinh cosh
cosh 2 = cosh2 + sinh2
sinh =1
2
cosh =1
2 +
They are derivativesof each other.
arctanh =1
2log
1 +
1
arctanh =
1
1 2
arcsinh =
1
2 + 1
arccosh =
1
2 1
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Riemann IntegrabilityConditions
Riemann Sum
Fundamental Theorem of Calculus LiebnizsRule
Odd & Even Functions Integration by Parts
Ordinary Differential Equationsof order
Separable ODEs
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=1
where are the boundaries of the subintervals, = 1 and , 1 . Labelled regular if all subintervals are equal length.
is Riemann integrable if upper bound of lower sums and lower bound of upper sums are equal. If:
a. continuous on [, ]
b. monotonic on [, ]
c. bounded on [, ] and removable discontinuities only
()
()
=
(for differentiating integrals with respect to a terminal variable)
If () is continuous for then
=
is differentiable for < < , () = () has right derivative () at = and left derivative () at = .
= () ==
Odd:
=
Even:
=
Those of the form
= ()
(Mentally go through procedure for solving.)
Those of the form
, ,
, ,
= 0
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General first-order linear ODEsand their integrating factors
First-order ODEs of homogeneous type
First-order ODEs of Bernoulli type Stability
Second-order linear ODEswith constant coefficients
Complementary Functionsof second-order linear ODEs
with constant coefficients
Strategies forfinding particular integrals
of second-order linear ODEswith constant coefficients
Comparison Testfor Improper Integrals
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Those of the form
=
(Mentally go through procedure for solving.)
Those of the form
+ =
with = exp () .
(Mentally go through procedure for solving.)
An equilibrium solution = of an ODE is stable if, given >0 > 0 such that for every othersolution () with 0 < , < (0,).
Asymptotically stable if
Those of the form
+ =
(Mentally go through procedure for solving.)
Solve characteristic equation.
If 2 solutions: 1 + 2
If 1 solution: ( + )1
(Go through complex root case mentally.)
Ones of the form
2
2+
+ =
(Mentally go through procedure for solving.)
If () is Riemann integrable on all finite subintervals of [,), 0 () () for , and the
improper integral ()
exists, then () exists and
0 ()
() .
Split up using superposition ideas.
Strategic guessing (go over strategies mentally).
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Limit Comparison Testfor Improper Integrals
The Gamma Function
Absolute Convergencefor improper integrals and series
Mean of aProbability Density Function
Fourier Transform The Harmonic Series
The Divergence Test(a.k.a. The Dont waste my time Test)
The Geometric Series(and its sum to infinity)
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= 0
1
! = + 1 = 0
If () is Riemann integrable on all finite subintervals of [,); () >0 for ; ()~() as and the improper integral
() exists, then
()
exists.
=
()
where () is the Probability Density Function.
If is Riemann integrable in all
finite intervals and
() 0 1, and ()
decays monotonically to zero as
, then =1 () converges
iff 1() < .
If > 0 and > 0 for all and ~ as then
either =1 and =1
both converge;
or =1 and =1
both diverge.
If > 0 and > 0 for all and = () as then
=1 converges =1
converges;
=1 diverges =1
diverges.
=1
1
Converges iff > 1.
If 0 and decays monotonically to zero as then the alternating series =0 (1) converges.
If > 0 for all , then
lim
+1
< 1 =1
converges;
lim
+1
> 1 =1
diverges.
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Multiplication of Series Power Series
Uniform Convergence of Series The Weierstrass M Test
The Taylor Series Pochhammers Symbol
Theorem on theConvergence of Power Series
Addition of Power Series
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Those of the form:
=0
0
centred on 0, being independent of .
If the series = = and =
= are both absolutely
convergent, then their product is given by the absolutely convergent series
= =
=
If () for all , where is independent of and =0 converges, then
=0 () converges uniformly on
the interval .
This sum is a continuous function if () is continuous.
If the series =0 () converges
for all where is an open or closed interval, we say that it converges uniformly on if the sequence of partial sums = =0 () converges uniformly on
.
Defined for {0} by
()
= 1, = 0
+ 1 + 1 ,
=0
()
!
The special case = 0 is sometimes called the Maclaurinseries.
If the power series =0
and =0
have radii of convergence and respectively, then =0
(+)
converges at least for