Free Essay

Submitted By gereksiz

Words 2166

Pages 9

Words 2166

Pages 9

1. Series

1.1 Numerical series

Definitions ➢ Let (an)n(1 a sequence of real numbers. The infinite sum (a1+a2+…+an+…) is called (numerical) series of general term an and is denoted by [pic] (or [pic]). ➢ The finite sum Sn= a1+a2+…+an=[pic], n(1 is named partial sum of order n, associated to the series[pic]. ➢ The series [pic] is convergent if and only if (iff) the sequence (Sn)n(1 is convergent (( S=[pic](R). In this case the limit S is named the sum of series[pic]. ➢ The series [pic] is divergent iff the sequence (Sn)n(1 is divergent (the limit [pic] is +(, -( or doesn’t exists).

Exercises

Study the nature (convergent or divergent) and if the case, compute the sum of the following series: a)[pic]; b) [pic](geometric series); c) [pic].

Properties

1) If in a series the order of a finite number of terms is changed then we get a series having the same nature as the first one.

2) If in a series a finite number of terms is added or subtracted the result is a series with the same nature as the first one.

3) The remainders of a convergent series, Rn= S-Sn form a sequence convergent to 0.

4) If the series [pic]is convergent then the sequence of partial sums (Sn)n(1 is bounded.

5) If the series [pic] and [pic]are convergent and have the sums A and, respectively B then the series [pic] is convergent and has the sum ((A+(B),( (, ((R.

6) If the series [pic]is convergent then[pic]=0.

Remarks

1) (Divergence criterion) If [pic]is a series such that [pic](0 then [pic] is divergent.

2) If [pic]=0 then the series [pic] may be convergent or divergent (doesn’t result that [pic]is convergent).

Exercises

Study the nature of the following series: a)[pic]; b) [pic]; c) [pic]; d) [pic]; e) [pic].

1.1.1. Alternating series

Definition

The series [pic] is named alternating if an(an+10 (n(1.

The convergence of an alternating series can be approached using the following criterion:

Leibniz criterion

Let [pic](or [pic]) an alternating series with an>0, (n(1. If 1) an(an+1, (n(1 (the sequence (an)n(1 is monotone nonincreasing); 2) [pic]=0. then the considered series is convergent.

Proof

Let the partial sums S2n=a1-a2+…+a2n-1-a2n and S2n+2=a1-a2+…+a2n-1-a2n+a2n+1-a2n+2. Then S2n+2- S2n= a2n+1-a2n+2(0 (using hypothesis 1)). So the sequence (S2n)n(1 is increasing (1).

In addition S2n can be written

S2n=a1-(a2-a3)-…-(a2n-2-a2n-1)-a2n,

but a2-a3(0,…, a2n-2-a2n-1(0 (by 1)) and a2n>0 (because an>0, (n(1) therefore we get S2n0, (n(1.

It is easy to demonstrate that, if [pic]is an (SPT), the sequence of partial sums (Sn)n(1 is increasing. In sequel, if we prove the sequence (Sn)n(1 is bounded then (Sn)n(1 is convergent, so the series[pic] is convergent.

Comparison criterion (I)

Consider two (SPT) [pic] and[pic]. If ( N(N* such that (n(N, an(bn then

1) [pic]convergent ([pic]convergent.

2) [pic]divergent ( [pic]divergent

Proof

1) If [pic]convergent then by Cauchy criterion (( ( N(()(N* such that (n(N((),(p(N*

|(bn+1+bn+2+…+bn+p(1.

Exercises

Study the nature of the series:

a) [pic]; b) [pic]; c) [pic]; d) [pic].

Root test (Cauchy)

Let a (SPT) series[pic].

1) If ( N(N and ( r((0, 1) such that [pic](r, (n(N then [pic]is convergent.

2) If we have [pic](1 for an infinite number of terms of sequence an then [pic]is divergent.

Proof

1) From hypothesis we have an(rn,(n(N with r((0, 1). In addition, for r((0, 1) the series[pic] is convergent. Therefore, by Comparison criterion (I) we get [pic]- convergent.

2) The second statement assumption implies that an(1 for an infinite number of terms from the sequence (an)n(1. So[pic](0 and, by Divergence criterion, is obtained[pic]is divergent.□

Corollary (Root test)

Let a (SPT) series[pic] and suppose that [pic] = l exists, then - if l>1 ( [pic]divergent - if l1such that [pic](r, (n(N, then [pic]is divergent.

Proof

1) For first part we have (n(N, [pic](r ( aN+1(raN, aN+2(raN+1(r2aN,…, aN+p (rpaN, (p(N*.

As the series [pic]is convergent (because r((0, 1)), then by applying Comparison criterion (I) is obtained that[pic]is convergent.

2) From the hypothesis results that (an)n(N is an increasing sequence and an>aN, (n(N. Therefore[pic](0 and by Divergence criterion results [pic]is divergent.

Corollary (Ratio test)

Let a (SPT) series[pic] and suppose that [pic] = l exists, then - if l>1 ( [pic]divergent - if l1 then [pic]is convergent.

2) If ( N(N such that (n(N, n([pic]-1) (r1 ( [pic]convergent - if l0.

Definition

A series [pic]is absolutely convergent if [pic]is convergent.

If a series [pic] is convergent and not absolutely convergent then [pic] is called conditionally convergent.

Remark

It is obvious that [pic] absolutely convergent ([pic]convergent but the reverse implication is not always true.

Exercises

Study the absolutely convergence and convergence of the series:

a) [pic]; b) [pic]; c) [pic]

1.2 Function series

Consider a set A(R and fn:A(R, n(1, n(N, a real function sequence, denoted (fn)n(1.

Definitions ➢ An element a(A is a convergence point of function sequence (fn)n(1 if the numerical sequence (fn(a))n(1 is convergent. The set D = {a(A((fn(a))n(1 is convergent} is named convergence set (domain) corresponding to the function sequence (fn)n(1. Next we define f:D(R, by f(x)=[pic]. Then we say the sequence (fn)n(1 is convergent on D to the function f; ➢ The sequence (fn)n(1 is simple convergent on A to the function f if (x(A, ((>0, (N((, x) (N such that (n( N((, x) we have (fn(x)-f(x)(< ( (notation: fn [pic]f);

Definitions ❖ Let a function sequence (fn)n(1 defined on a set A. The (infinite) sum f1+f2+…+fn+… is called function series and is denoted by[pic]. ❖ The function sequence (Sn)n(1 given by Sn=[pic] is named the partial sum associated to the function series [pic]. ❖ We say the function series [pic]is convergent at a(A if (numerical series) [pic] is convergent (that is, the function sequence (Sn)n(1 is convergent at x=a). ❖ The function series [pic]is absolutely convergent at a(A if [pic] is absolutely convergent. ❖ The set D = {a(A( [pic]is convergent at a} is called the convergence set (domain) of function series[pic].

1.2.1. Power series

Consider a set B(A and f:B(R. ❖ The function series [pic] is called power series defined on R if (n(N, (x(R, fn(x)=anxn, where an(R (notation:[pic])

Remark

The power series [pic]is convergent at x=0 and the sum equals a0.

The convergence domain for a power series can be expressed in a convenient way. For this purpose we give here (without proof) Abel theorem.

Theorem 1 (Abel)

Let a power series[pic]. There is a real number R(0 finite or infinite, called convergence radius, such that:

1) [pic]is absolutely convergent on (-R, R);

2) [pic]is divergent on (-(, -R)((R, ().

Remarks

• If D = {a(A([pic] is convergent at a} is the set (domain) of convergence for the power series[pic], then by Abel theorem we get (-R, R)(D( [-R, R]. • The power series convergence for x=(R is studied using criteria from numerical series theory.

Next, we are interested in obtaining a formula to compute the convergence radius of a power series. The Cauchy-Hadamard theorem gives a useful result for this problem.

Theorem 2 (Cauchy-Hadamard) Consider the power series [pic]and R its convergence radius. If the following limit (=[pic] ([0, (] exists, then R=[pic].

Remark

Sometimes is easier to compute ( as: (=[pic] (if the limit exists).

Exercise

Determine the convergence domains for the following power series:

a) [pic]; b) [pic]; c) [pic]; d) [pic]

2. Functions of several variables

Consider Rn={x=(x1, x2, …, xn)(xi(R, i=[pic]}.

Definition

The function d: Rn( Rn([0, () is said to be a distance if:

1) d(x, y)(0, (x, y( Rn and d(x, y)=0(x=y.

2) d(x, y)=d(y, x), (x, y( Rn;

3) d(x, z)(d(x, y)+d(y, z), (x, y, z( Rn.

Example

The function d: Rn( Rn([0, (), defined by d(x, y)=[pic]for x=(x1, x2, …, xn) and y=(y1, y2, …, yn) is a distance. If n=1 then d(x, y)=(x1-y1(, where x=x1 and y=y1. For n=2, x=(x1, x2), y=(y1, y2) we have d(x, y)= [pic].

Definition

Consider x0(Rn and r>0. Then the set Sr(x0)={x(Rn(d(x, x0)0 such that Sr(x0) (V.

Proposition

Any open sphere centered at x0 contains an n-dimensional interval which includes x0 and reciprocally.

2.1 Continuity

Definition (limit of a function) ▫ x0(Rn is an interior point to the set A( Rn iff (r>0 such as Sr(x0) ( A.

In the following definitions we take n=2, but the notions can be easily extended to n>2.

Consider A( R2, f: A(R and (a, b) an interior point to A. ▫ We say [pic]= l([pic] iff ({(xn, yn)}n(1(A with (xn, yn)[pic] (a, b) and (xn, yn)((a, b) (n(1 we have [pic]=l.

Remark

An equivalent definition for the limit of a function in a point is the following:

[pic]= l(R iff ((>0, ((=((()>0 such as ((x, y) (A with (x-a(0 such that [pic],[pic],[pic], [pic] are continuous on Sr(a, b). Let the (Hessian) matrix

H(a, b)=[pic]and (1=[pic], (2=det H(a, b).

If (2>0 then (a, b) is a local extreme point as follows: • if (1>0 then (a, b) is a local minimum; • if (10, (2>0,…, (n>0 and negative defined iff (10,…, (-1)n(n>0 (alternating signs). So: - if H(a) is negative defined then x=a is a local maximum. - if H(a) is positive defined then x=a is a local minimum.

2.4 Application in economics

2.4.1. Marginal value and partial elasticity of functions

Consider a set A(Rn and a function f: A(R that admits first order partial derivatives on A, [pic]: A(R, ( i=[pic]. We define ➢ Marginal value of f with respect to xi at point (x1, x2, …, xn) as [pic]=[pic] ➢ Rate of variation of f with respect to xi at point (x1, x2, …, xn) as

[pic]=[pic]

➢ Partial elasticity of f with respect to xi as

[pic]=[pic].

Example

Assume n goods X1, …, Xn are sold on a market at the prices p1, …, pn. In this case the demand of Xi, denoted Yi, is a function depending on the prices p1, …, pn:

Yi=fi (p1, …, pn), i=[pic].

We suppose that (i=[pic], fi admits first order partial derivatives with respect to pj, (j= [pic] (([pic](i, j= [pic]). So, the marginal value [pic]=[pic]shows the speed of Yi demand decreasing corresponding to a pj price increasing (usually [pic]0, [pic]>0 the goods are called substitutes (the demand of Xi increases when the price pj of Xj increases and conversely).

If [pic]0.

Remark

The gamma integral is convergent for every p>0.

Properties

The gamma integral has the following properties: 1) ((1)=1; 2) ((p)=(p-1) ((p-1), p>1; 3) ((n)=(n-1)!, (n(N; 4) (([pic])=[pic].; 5) ((a)( ((b)=[pic], if a+b=1, a, b>0.

Exercise

Prove the properties 1-3.

3.2. Beta integral

The improper integral [pic], p, q>0 is called beta integral (beta function) and is denoted ((p, q)= [pic].

Remark

The beta integral is convergent for every p, q>0.

Properties

The beta integral has the following properties: 1) ((p, q)= ((q, p), (p, q>0; 2) ((p, q)=[pic] ((p-1, q), (p>1, q>0; 3) ((p, q)= [pic]((p, q-1), (p>0, q>1; 4) ((p, q)= [pic]((p-1, q-1), (p, q>1; 5) ((p, q)=[pic], (p, q>0.

3.3. Euler -Poisson integral

The improper integral [pic]is called Euler-Poisson integral.

Proposition

The Euler-Poisson integral equals [pic].

Proof

In the Euler-Poisson integral is considered the notation x2=t (dx=[pic]dt) so we get

[pic]=[pic]=[pic]=[pic](([pic])=[pic].

[pic]