Explore BrainMass

Explore BrainMass

    Real Analysis

    BrainMass Solutions Available for Instant Download

    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 : Measurable Sets and Functions

    1).If f: X--> C ( C is complex plane) is measurable, then prove that f^-1({0}) ( f inverse of 0 or any other point) is a measurable set in X. 2). If E is measurable set in X and if X_E ( x) = { 1 if x is in E, 0 if x is not in E} then X_E is a measurable function. Now I want you to prove the other direction, that is, I w

    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?

    Evaluating a limit

    Please evaluate that attached limit... (See attached file for full problem description)

    Limits : Application to Relativistic Mass

    In the theory of relativity, the mass of a particle with velocity v is m = m0/sqrt(1- v^2/c^2) where m0 is the rest mass of the particle and c is the speed of light. What happens as v ?> c-?

    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.

    Epsilon-delta method of limits

    Use the formal definition of the limit (epsilon-delta method) to show that: (See attached file for equation) Please be very thorough in your explanation of the solution by showing each step in detail. I really want to understand the method and be able to apply it to other problems. Thanks in advance! --- (See attach

    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?

    Taylor Series Representation and Residue

    Write f(z):=16z/(z^2 +1)^3 as f(z)= h(z)/(z - i)^3 with an explicit expression for the function h(z). Explain why h(z) has a Taylor series representation about i and use this representation to find explicitly the principal part off at i. Hence, find the numerical value of the residue of f at i. Please see the at

    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∙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

    Integer Sum of Series

    Determine the sum of the integers among the first 1000 positive integers which are not divisible by 4 or are not divisible by 9. (This is not an exclusive or)

    Finding a Function that Satisfies a Limit

    Find a formula for a function f, that satisfies the following conditions: 1. lim(x->+/-infinity)f(x) = 0, 2. lim(x->0)f(x) = -infinity, 3. lim(x->3-)f(x) = infinity, 4. lim(x->3+)f(x) = -infinity, 5. f(2) = 0.

    Infinite Series : Convergence and Divergence (8 Problems)

    1) a) Prove that N ∑ 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