Principles Of Mathematical Analysis. Chapters 3, 4,5 Rudin

Front 1

Upper/Lower Limits

Back 1

Let {sn} be a sequence of real numbers. Let E be the set of numbers x (in the extened real number system) such that snk ==>x for some subsequence {snk}. This set E contains all subsequential limits plus possibly +∞ and -∞. s*=sup E and s*=inf E. The numbers s*, s* are called the upper/lower limits of {sn}. lim sup sn= s* and lim inf sn= s*.

Front 2

Series

Back 2

The definition given here will be for real numbers, but can be generalized.
Given an infinite sequence of real numbers {an}, define
SN= ∑ an = a0 + a1 + … + aN.
Call SN the partial sum to N of the sequence {an}, or partial sum of the series. A series is the sequence of partial sums, {SN}.

Front 3

Convergent/Divergent Series

Back 3

A series  ∑an  is said to 'converge' or to 'be convergent' when the sequence {SN} of partial sums has a finite limit. If the limit of SN is infinite or does not exist, the series is said to diverge. When the limit of partial sums exists, it is called the sum of the series.
∑an converges if and only if for every ε>0 there is an integer such that |∑ak|≤ε m≥n≥N.
If ∑an converges, then lim an=0(n==>∞)

Front 4

Geometric Series

Back 4

A geometric series is one where each successive term is produced by multiplying the previous term by a constant number. Example: if 0≤x≤1 then ∑xn=1(/1-x) if x≥1 then the series diverges.

Front 5

Root Test

Back 5

Given ∑an, put α=lim sup (|an|)1/n:
a)if α<1, ∑an converges
b) if α>1, ∑an diverges
c) if α=1, the test gives no information

Front 6

Ratio Test

Back 6

The series ∑an converges if lim sup |(an+1)/an|<1 and diverges if |(an+1)/an|≥1 for n≥n0 where n0 is an integer. Note: being equal to one says nothing about convergence.

Front 7

Power Series

Back 7

Given a sequence {cn} of complex numbers, the series ∑cnzn is called a power series. The numbers cn are called the coefficients of the series;z is a complex number

Front 8

Radius of Convergence

Back 8

Given the power series ∑cnzn ,put α=lim sup (|cn|)1/n and R=1/α
(If α=0,R=+∞;if α= +∞,R=0) Then ∑cnzn converges if |z|<R and diverges if |z|>R. R is the radius of Convergence

Front 9

Absolute Convergence

Back 9

The series ∑an is said to converge absolutely if the series ∑|an| converges. If ∑an converges absolutely , then ∑an converges. Proof by Cauchy and triangle thm.

Front 10

Cauchy Product

Back 10

Given ∑an, ∑bn we put cn=∑akbn-k==∑bkan-k and call ∑cn the product of the two given series.

Front 11

Rearrangements

Back 11

Let {kn},n =1,2,3,…, be a sequence in which every positive integer appears once and only once(that is, {kn} is a 1-1 function from J to J. Putting a’n=akn (n=1,2,3,…) we say that ∑a’n is a rearrangement of ∑an .

Front 12

Limits of a Function

Back 12

Let X and Y be metric spaces ; suppose E ⊂ X f maps E into Y , and p is a limit point of E. We write f(x)==>y as x==>p or lim f(x)= q as(x==>p); if there is a point q ∈ Y with the follow property: For every ε>0 there exists δ>0 such that dY(f(x),q)< ε for all points x ∈ E for which 0< dX(x,p)< δ

Front 13

Addition/Multiplication of Functions

Back 13

Suppose E ⊂ X, a metric space, p is a limit point of E, f and g are complex functions of E and lim f(x)=A and lim g(x)=B as x==>p. Then a) lim (f+g)(x)=A+B b) lim (fg)(x)=AB c)lim(f/g)(x)= A/B if B≠0.

Front 14

Continuous functions

Back 14

Suppose X and Y are metric spaces, E⊂ X, p ∈ E and f maps E into Y. Then f is said to be continuous at p if for every ε>0 there exists a δ>0 such that dY(f(x),f(p))< ε for all points x ∈ E for which dX(x,p)< δ. If f is continuous at every point of E, then f is said to be continuous on E.

Front 15

Bounded Functions

Back 15

A mapping of f of a set E into Rk is said to be bounded if there is a real number M such that |f(x)|≤M for all x ∈ E

Front 16

Uniformly Continuous

Back 16

Let f be a mapping of metric space X into metric space Y. We say that f is uniformly continuous on X if for every ε>0 there exists a δ>0 such that dY(f(q),f(p))< ε for all points q,p ∈ X for which dX(q,p)< δ.

Front 17

1st and 2nd type discontinuities

Back 17

Let f be defined on (a,b). If f is discontinuous at a point x, and if f(x+) and f(x-) exists, then f is said to have a discontinuity of the first kind, or a simple discontinuity at x. Otherwise the discontinuity is said to be of the second kind.

Front 18

One sided limits

Back 18

In calculus, a one-sided limit is either of the two limits of a function f(x) of a real variable x as x approaches a specified point either from below or from above.

Front 19

Monotonic Functions

Back 19

Let f be real on (a,b). Then f is said the be monotonically increasing on (a,b) if a<x<y<b implies f(x)≤f(y). If the last inequality reversed we obtain the definition of a monotonically decreasing function.

Front 20

Derivative of a Function

Back 20

Let f be defined ( and real-valued) on [a,b]. For any x ∈[a,b] form the quotient φ(t)=(f(t)-f(x))/(t-x) (a<t<b,t≠x) and define f’(x)=lim φ(t) (t==>x) .
If f’ is defined at a point x we say that f is differentiable at x. If f’ is defined at every point of a set E⊂[a,b] we say that f is differentiable on E.

Front 21

Local Maximums and minimums

Back 21

Let f be a real function defined on the metric space X. We say that f has a local maximum at a point p ∈ X if there exists δ>0 such that f(q)≤f(q) for all q ∈ X with d(p,q)< δ. Local minima are defined similar with a reverse of the relation.

Front 22

Higher Derivatives

Back 22

If f has a derivative f’ on an interval and if f’ is itself differentiable, we denot the derivate of f’ as f’’ and call f’’ the second derivative of f. In order for f(n)(x) to exist at a point x, f(n-1)(t) must exist in a neighborhood of x and f(n-1) must be differentiable at x.

Front 23

Taylor’s Theorem.

Back 23

Suppose f is a real function on [a,b], n is a positive integer, f(n-1) is continuous on [a,b]. f(n)(t) exists for every t ∈ (a,b). Let α,β be distinct point of [a,b] and define P(t)=∑(f(k)(α)/k!(t-α)k. Then there exists a point x between α and β such that f(β)=P(β)+f(n)(x)/n!(β-α)n. For n=1 this is just the mean value theorem. In general the theorem shows that f can be approximated by a polynomial of degree n-1.

Front 24

Partition

Back 24

Let [a.b] be given interval. By a partition P of [a,b] we mean a finite set of points xo,x1,...xn, where a=x0⪯x1⪯...⪯xn-1⪯xn=b. We write ⨺xi=xi-xi-1 (i=1,...,n) .
Now suppose f is bounded real function defined on [a,b]. Corresponding to each partition P of [a,b] we put Mi=supf(x); mi=inf f(x) for (xi-1⪯x⪯xi). Then U(P,f)=ΣMi⨺xi and L(P,f)=Σmi⨺xi

Front 25

Upper/Lower Riemann Integrals

Back 25

⨛fdx=inf U(P,f) is the upper Riemann Integral
⨜fdx=sup L(P,f) is the lower Riemann intergral

Front 26

Upper/Lower Stieltjes Integral

Back 26

Let α be a monotonically increasing function on [a,b] corresponding to each partition of [a,b], we write ⨺αi=α(xi)-α(xi-1). It is clear that ⨺α≥0. For any real function f which is bounded on [a,b] we put U(P,f,α)=ΣMi⨺αi and L(P,f,α)=Σmi⨺αi where Mi, and mi have the same meaning as Riemman Integrals. Then the upper Stieltjes Integral is ⨛fdα=inf U(P,f,α) and the Lower Integral is ⨜fdα=inf L(P,f,α)

Front 27

Refinement /Common Refinement

Back 27

We say that the partition P* is a refinement of P if P*⊃P that is if every point of P is a point of P*). Given two partitions, P1 and P2, we say that P* is there common refinement if P*=P1 U P2.

Front 28

Riemann Integral

Back 28

When

⨛fdx=inf U(P,f) equals ⨜fdx=sup L(P,f) we say is Riemann Integral of [a,b] and say
∫f(x)dx.

Front 29

Riemann-Stieltjes Integral

Back 29

When the Upper Integral⨛fdα=inf U(P,f,α) and the Lower Integral ⨜fdα=inf L(P,f,α) are equal we denote
∫f(x)dα(x) or ∫fdα

When α(x)=x then the integral becomes the Riemann Integral

Front 30

Unit Step Function

Back 30

The unit step function I is defined by
I(x)=
0 when x⪯0
1 when x>0