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Real Analysis

Mapping, Contraction and Fixed-Point Theorem

(See attached file for full problem description with proper equations) --- 3. Let T(x) = x^2 Show that T is a contraction on (0, 1/3] , but that T has no fixed point on this interval. Does this conflict Theorem 6.4? Explain. Note: We are using the book Methods of Real Analysis by Richard R. Goldberg. This

Fibonacci Sequences, Convergence and Limits

(a) find the first 12 terms of the Fibonacci sequence Fn defined by the Fibonacci relationship Fn=Fn-1+Fn-2 where F1=1, F2=1. (b) Show that the ratio of successive F's appears to converge to a number satisfying r2=r+1. (c) Let r satisfy r2=r+1. Show that the sequence sn=Arn, where A is any constant, satisfies the Fi

Increasing, Bounded, Sequences and Series, Limits and Convergence

The sequence Sn = ((1+ (1/n))^n converges, and its limit can be used to define e. a) For a fixed integer n>0, let f(x) = (n+1)xn - nxn+1 . For x >1, show f is decreasing and that f(x) . Hence, for x >1; Xn(n+1-nx) < 1 b) Substitute the following x-value into the inequality from part (a)

Jacobians of the Algebraic Functions

Independence and relations Real Analysis Jacobians (XI) If u = (y^2)/(2x) and v = (x^2 + y^2)/(2x), find the Jacobian of u,v with respect to x,y. The fully formatted problem is in the attached file.

Real Analysis - Limits of Sequences

Suppose that {an} and {bn} are sequences of positive terms, and that the limit as n goes to infinity of (an/bn) = L > 0. Prove that limit as n goes to infinity of an is positive infinity if and only if the limit as n goes to infinitiy of bn is positive infinity. Here is what I have for proving the first way: Suppose that

Real Analysis: Elementary Sets and Closure

1) Let M be an elementary set. Prove that | closure(M)M | = 0. (closure of M can also be written as M bar, and it is the union of M and limit points of M). 2) If M and N are elementary sets then show that | M union N | + | M intersection N| = |M| + |N| The definition of elementary set : If M is a union of finite members

Real Analysis lim sup and lim inf

Let {a_n} and {b_n} be sequences in [-infinity,+infinity] and prove the following assertions: 1). a).Lim sup (as n -> infinity) ( a_n + b_n) less than or equal to lim sup a_n + lim sup b_n ( as n foes to infinity). b).Show by an example that strict inequality can hold. Provided none of the sums is of the form infin

Radius of convergence in series (complex plane)

1). Find the radius of convergence for each of the following power series. Please check my solution for this problem: a). sum ( n = 0 to infinity) a^n z^n, a is a complex number. My solution: R( radius of convergence) = lim |a_n/a_n+1) = lim | a^n/a^(n+1)| = 1/|a| b). Sum ( n=0 to infinity) = lim|a^(n^2)*z^n, a is

Real Analysis Topology and Sigma-Algebra

1). Prove that any sigma-algebra, which contains a finite number of members is also a topology. ( The Q in another words : to show that there exist a sequence of disjoint members of a sigma algebra which contains infinite no. of members). 2). Does there exist an infinite sigma-algebra which has only countably many members?

Real Analysis : Sigma-algebra

Suppose X is a measurable space, E belongs to the sigma algebra ( I believe to the sigma algebra in X) , let us consider XE = Y. Show that all sets B which can be expressed as AE, where A belongs to the sigma algebra in X, form a sigma-algebra in Y. Please justify every step and claim you make in the solution.

Effects of Inflation and Deflation and Types of Annuities

1) How would economic pressures like inflation or deflation affect your decision to make a long term investment? Should our mathematical analysis take these factors into consideration? 2) What are some different types of annuities that you have used or heard of?

Finding the Sum of a Harmonic Series

The series .... 1 + 1/2 + 1/3 + 1/4 + 1/6 + 1/8 + 1/9 + 1/12 + .... Where the terms are the reciprocals of the positive integers whose only prime factors are 2's and 3's. What is the sum of the series?

Real Analysis : Limits and Cluster Points

Please see the attached file for the fully formatted problems. 1) Prove that does not exist but that . 2) Let f, g be defined on to , and let c be a cluster point of A. Suppose that f is bounded on a neighborhood of c and that . Prove that . 3) Let f, g be defined on A to and let c be a cluster point o

Taylor Polynomial

Use Taylor polynomials about 0 to evaluate sin(0.3) to 4dp,showing all workings. 1)F(x)=square root 4+x and G(x)=square root 1+x by writing square root of 4+x=2 square root 1+1/4x and using substitution in one of the standard Taylor series, find the Taylor series about 0 for f.Given explicitly all terms up to term in x raise

Sequences and limit boundaries

1) If (bn) is a bounded sequence and lim(an) = 0, show that the lim(anbn) = 0. Explain why Theorem 3.2.3 cannot be used. Note: Here's Theorem 3.2.3 (a) Let X = (xn) and Y = (yn) be sequences of real numbers that converge to x and y respectively, and let c be an element R. Then the sequences X+Y, X-Y, X&#8729;Y, and cX co

Project Evaluation Review Technique and Critical Path Method

Please refer to the attached file for this PERT / CPM problem: I've determined that the critical path for this network is A - E - F and the project completion time is 22 weeks. Here's where I need help: If a deadline of 17 weeks is imposed, what activities should be crashed?

Pin-Joint Analysis : Direction and Magnitudes of Support Reactions

I need a worked solution for these questions (also attached on the LAST PAGE of the attachement. The first couple pages are just examples. Thanks) The structure shown in Figure TA 1 is a pin-jointed section of a canopy and carries a single load of 4 kN acting at the lower right-hand joint. [DIAGRAM] Pin-jointed canopy De

Infinite Series : Convergence and Divergence (8 Problems)

1) a) Prove that N &#8721; 1/n(n+1) = 1- (1/N+1) n=1 Hence, or otherwise, determine whether the following infinite series is convergent or divergent: b) Determine whether each of these infinite series are convergent or divergent. Justify your an

Taylor Expansion, Interval of Convergence and Calculation of 'pi'

In 1671, James Gregory, a Scottish mathematician, developed the following series for tan^-1 x {See attachment} 1. Verify that Gregory's series is correct by using a Taylor Series expansion or methods of power series. 2. Find the interval of convergence of Gregory's series. 3. Using Gregory's series, find a series whose

Real Analysis : Continuity, Closed and Open Sets and Differentiability

Prove OR disprove the following statements. Explain. (i) There is a nonempty set S in R ( real numbers) such that S contains none of its limit-point(s). (ii) There is a nonempty set S in R such that S is neither open nor closed. (iii) There is a nonempty set S in R such that S is both open and closed. (iv) Let a

Real Analysis : Differentiability and Sequence of Partial Sums

A). Prove that the function f(x) = e^x is differentiable on R, and that (e^x)' = e^x. ( Hint: Use the definition of e^x, and consider the sequence of partial sums.) My thoughts on a: I tried to prove the differentiability by proving continuity on R, since e^x is series, sum of polynomials, and all polynomials are different

Closure, Convergence, Differentiability, Integrability, Sequences

Prove or disprove the following statement. Provide detailed answer and justify all steps. 1). There is a nonempty set S in R such that closure of S is equal to R and the closure of its complement closure(R-S) also is equal to R. My thoughts on this problem: Q ( rationals) and R-Q (irrationals), but how to prove that the cl