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Real Analysis

Contradiction Real Analysis is Performed.

Please see the attached file. We call an extended real number a cluster point of a sequence if a subsequence converges to this extended real number. Show that lim inf(a_n) is the smallest cluster point o (a_n and lim sup(a_n) is the largest cluster point of (a_n).

Locating the limit of the sequence

A bored student enters the number 0.5 in her calculator, and then repeatedly computes the square of the number in the display. Taking a_0 = 0.5, find a formula for the general term of the sequence {a_n} of the numbers that appear in the display, and find the limit of the sequence {a_n}.

Analysis: Taylor Series, Radius of Convergence

Let f(x)=ln?(1+x). (a) Prove by induction that for n ≥ 1, f^(n) (x) = (-1)^(n-1) ((n-1)!/(1+x)^n )) (It is not necessary here to determine the derivatives from first principles) (b) Write down the Taylor series for f at a = 0. (c) What is the radius of convergence of the power series in (b)? (d) Use Taylor p

Determining a Quadratic using the Taylor Series

Let G(x,y)= ln(1+xy). Determine a quadratic in x and y that is a good approximation to G(x,y) for points that are close to (0,0). Use your answer to approximate G(0.05, -0.02) and compare the approximation with the function value. (Expand x and y till powers of 2)

Estimating Population Mean: Central Limit Theorem

If we increase the sample size, n, how does that increased sample size affect our ability to estimate the population mean? How does the population's shape affect the estimate of the mean? Why do so many of life's events share the same characteristics of the Central Limit Theorem?

The Limit of this Sequence - Bounded Measurable

Show that if A is bounded, then A is measurable if and only if A intersect [-n,n] is measurable for every positive integer n, and in that case m(A)=lim (A intersect [-n,n]). See the attached file.

Pointwise Limits

Determine the pointwise limit of (f_n), then decide whether the convergences is uniform or not (see attachment). 1. f_n(x)= x/n, x is all real number 2. f_n(x) = (sin(n*x))/n*x, where 0<=x<=1 3. f_n(x)=(x^n)*(1-x^n), where 0<=x<=1 4. f_n(x)={nx, for 0<=x<=1/n and 0, for 1/n<x<=1.

Current in Series, Resistance in Parallel Circuit, Power

Please break these examples down for me so I can grasp this before testing. Total Current in series circuit with following: Vs=20V, R1=3k ohms, R2 = 2k ohms, R3= 1k ohms Total equivalent resistance in a parallel circuit: Vs = 20V, R1=3k ohms, R2= 2k ohms, R3 = 1k ohms...all resistors are parallel. Total Current in par

Uniform Limit of a Sequence of Functions

Prove that any continuous function g :[0,pi}->Reals is a uniform limit of a sequence of functions g_n(x)= b_0 +b_1 cos(x) + ...+b_n cos(nx). I think Stone's theorem is needed here. Here is the version of Stone's theorem for the Reals. Let f:[a,b]-> Real be continuous. Let A be an algebra of functions defined over [a,b] suc

Problems with Antiderivatives

Determine the value of a that makes F(x) an antiderivative of f(x) f(x)=9square root x , F(x)=ax^3/2 Please provide any tricks I can use to remember steps to these kinds of problems. Find antiderivatives of the given function. f(x)=4/3x^1/3 Please show step by step and example any rules that need to be remembered.

Limits and uniform convergence

Consider the sequence of functions f_n(x) =sin ([2npi)^2+x]^(1/2)) on [0,infinity) a. Show that lim( n goes to infinity) f_n(x)=0 Hint: Use the mean value theorem for f_n(x) on [0,x] b. Show that f_n(x) converges uniformly to 0 on [0,a] for a fixed a in [0,infinity). c. Is it true that f_n(x) converges uniformly to

Limits, convergence and divergence

Suppose a_n is strictly greater than zero a. Show that lim( as n goes to infinity) a_n/(a_n+1) =0 if and only if lim (as n goes to infinity) a_n=0 b, Prove that sum (n = 1 to infinity) a_n diverges if and only if sum(n=1 to infinity) a_n/(a_n+1) diverges. c. What can be said about (convergence or divergence) of su

Interval and Radius of Convergence

Please see the attached image for complete questions. 1. Find the interval of convergence (including a check of end-points) for each of the given power series. 2. Use the geometric series test (GST) to write each of the given functions as a power series centred at x=a, and state for what values of x the series converges.

Sequences and Limits on Epsilon

See the attached file. 1.Let A be subset of R^p and x belongs to R^p. Then x is a boundary point of A if and only if there is a sequence {a_n} of elements in A and a sequence of {b_n}elements of C(A) (complement of A) such that lim(a_n)=lim(b_n). 2.Prove if{x_n} (n=1..infinity) is a sequence in R^p, x in R^p then TFAE:

Reorder Point Inventory Analysis Problem

Using excel. Tracy McCoy is the office administrator for the department of management science at Tech. The faculty uses a lot of printer paper, although Tracy is constantly reordering, paper frequently runs out. She orders the paper from the university central stores. several faculty members have determined that the lead

Power Series Problem and Solution

Please help me with the following question : Find a power series solution in power of x. 1) y" â?" 3y' + 2y = 0 2) x y" + (1-2x) y' + (x-1) y = 0 3) 2x(x-1) y" â?" (x+1) y' + y = 0

Convergence and divergence of series..

1. If a and b are positive integers then show that sum (n=1 to infinity) (1/(an+b)^p) converges for p greater than 1 and diverges for p less than or equal to one. 2. Let a be greater than zero. show that the series sum (n=1 to infinity)(1/(1+a^n)) is divergent if a is greater than zero or less than or equal to one, and

Convergence or Divergence of the Series

For the following either prove it or give a counterexample for 1 and 2 1. If sum(a_n) with a_n>0 is convergent. then is sum([a_n] ^2) always convergent. 2. If sum(a_n) with a_n>0 is convergent. then is sum([a_n a_(n+1)] ^0.5] always convergent. 3. If sum(a_n) with a_n>0 is convergent. and b_n= (a_1 + ...+ a_n)/n for n i

Power Series Representation for the Function

Find a power series representation for the function and determine the interval of convergence: F(x) = (1 + x) / (1 - x) __________________________________ (1 + x) times the sum from n=0 to infinity of x ^n = the sum from n=0 to infinity of x ^n + x times the sum from n=0 to infinity of x ^n = the sum from n=0 to infin

Convergence a la Central Limit Theorem

We are given a sequence (X_k) of independent r.v.s with cumulative distribution functions (F_k). The cdfs are all continuous and strictly increasing. Show that the sequence of Z_n = -1/sqrt(n) * sum_{k=1}^n [ 1 + log(1 - F_k (X_k)) ] for n = 1, 2, 3, ... converges in distribution to a standard normal distribution.

radius and interval of convergence of each power series

Please see attached problems. [4] Find the interval of convergence for the following geometric series and, within this interval, find the sum : [5] Find the radius and interval of convergence of each power series. (a) (b)

Element of Infinitely Many Subsets of a Given Set

Please see the attached file. Let A_n be a sequence of subsets of a given set X, and let J be the set of all x in X that are in infinitely many A_n. Show that J is equal to the intersection (from n = 1 to infinity) of [the union (from i = n to infinity) of A_i].

Analysis - uniform convergence

For each natural number n and each number x in (-1,1), define f_n (x)=&#8730;(x^2+1/n) and define f(x)=|x|. Prove that the sequence {f_n} converges uniformly on the open interval (-1,1) to the function f. Check that each function f_n is continuously differentiable, whereas the limit function f is not differentiable at x=0.

radius of the power series

Find the radius of convergence of ∑▒〖(a_n x^n)〗, given a_n=1when n is the square of a natural number and a_n otherwise. If a_n=1 when n=m! for n∈N and a_n=0 otherwise, find the radius of convergence of ∑▒〖(a_n x^n)〗. Determine the radius of convergence of ∑▒〖(a_n x^n)〗, if 0<ρ≤|a_n |≤q for a

uniform convergence and proof

Determine the uniform convergence and convergence of the series ∑▒〖〖(f〗_n),〗 where f_n (x)is given by the following: (The Weoerstrass M-Test will be needed) a sin⁡(x/n^2 ) b. 〖(nx)〗^(-2),x≠0, c. 〖(x^2+n^2)〗^(-1) d. (-1)^n (n+x)^(-1),x≥0, e. 〖(x^n+1)〗^(-1),x≥0

Cost Accounting CVP Analysis and Price Changes

Scholes Systems supplies a particular type of office chair to large retailers. Scholes is concerned about the possible effect of inflation on its operations. Presently, the company sells 80,000 units for $60 per unit. The variable production costs are $30, and fixed costs amount to $1,400,000. Production engineers have advised m

convergence or divergence infinite series

Discuss convergence or divergence of the series whose nth term is 〖(-1)〗^n n^n/((〖n+1)〗^(n+1) ) (b) 〖(-1)〗^n 〖(n+1)〗^n/n^n (c) n^n/((〖n+1)〗^(n+1) ) (d) 〖(n+1)〗^n/n^(n+1) Given that ∑▒a_n is a convergent series of real numbers, Prove

Determine if Sequences of Functions Converges Pointwise: Example

Determine whether each of the following sequences of functions converges pointwise. If so determine if the convergence is uniform. See attachment for clarity. (i.) fn(x) =9 for x E [0, 1]. (ii.) fn(x) = 9(1-x) for x [0, 1]. (iii.) fn(x) = 9(1- 9) for x [0, 1].

Maclaurin series definition for series representation

1. Find the maclaurin series for f(x) = e^(3x) using the definition of the Maclaurin series. Assume that f has a power series representation. Find the associated radius of convergence. 2. Find the taylor series for f(x) = sin(x) centered at a = (pi/2), using the definition of a taylor series. Assume that f has a power series