### 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.

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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.

Find Limits and Rate of Convergence.

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

------------------------------------------------------------------------------------------- 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

Prove that if lim(|c_(n+1)/c_n|) = a>0 then lim(|c_n|^1/n) = a

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

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)

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-?

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.

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

Evaluate the following limit: lim (x-8) divided by, (the cubed root of x, minus 2) x->8

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?

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

Please see the attached file for the fully formatted problems.

Find the limit if it converges if it does not diverge an = 1/n^2 + 2/n^2 + 3/n^2 + ...... n/n^2 does it diverge? If it converges whats the limit?

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

Determine if the summation series is convergent or divergent from n=2 to infinity of 1/n(ln(n))^2.

Please see the attached file for the fully formatted problem.

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

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

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?

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

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)

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.

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

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

1. Determine whether or not the alternating series converge or diverge.... Please see the attached file for the fully formatted problems.

Let I:=[a,b] be a closed bounded interval and let f:I->R be continuous on I. Then f has an absolute maximum and an absolute minimum on I.

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