Triangle Inequality Definition of Convergence of

Sequences

Bolzano Weierstrass Theorem Cauchy Sequences

Contractive Sequences Order Hierarchy

Surjection Injection

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.

Definition of Convergence of Functions at a point

= ?

=

= ?

=

= ?

+

= ?

Continuity Uniform Convergence of Functions

Intermediate Value Theorem Differentiability

Rolles Theorem Mean Value Theorem

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.

LHpitalsRule Taylor Polynomial

Lagranges Remaindersin(y)

cos(y)

sinh(y), cosh(y)and their derivatives

Other usefulHyperbolic Function Identities

arctanh(t)and its derivative

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

Riemann IntegrabilityConditions

Riemann Sum

Fundamental Theorem of Calculus LiebnizsRule

Odd & Even Functions Integration by Parts

Ordinary Differential Equationsof order

Separable ODEs

=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

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

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).

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)

= 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.

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

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