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Real Analysis

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

Uniform Convergence of Series

If |fn(x)| < gn(x) for all nE R and every x E[a,b] , and the series... converges uniformly in [a,b], then ... converges uniformly in [a,b]. Please see the attached file for the fully formatted problems.

Limits and Convergence of Sequence

Prove that the sequence of functions ... converges for every , and find the limit to which it converges. Please see the attached file for the fully formatted problems.

Topology proof

Suppose that f:X->Y is continuous.... --- (See attached file for full problem description)

Real Analysis, sup, inf, measurable functions.

------------------------------------------------------------------------------------------- 1). If g_n = Sup f_n, then prove that ( g_n)^-1 ( ( alpha, infinity] ) = union ( n = 1 to infinity) (f_n)^-1((alpha,infinity]). ------------------------------------------------------------------------------------------- 2). Pr

Find the radius of convergence for each of the following power series

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


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.

Proving A Limit Using Epsilon-Delta Method

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 of f at i. Hence, find the numerical value of the residue of f at i. Please see t

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?

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 Limits

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 &#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