These problems are real analysis related problems. One of these is Liptuaz continuity/holder condition. It would be nice if you use mean value theorem for solving that problem. 1. Let f : [0, 2] ? R be continuous, assume that f is twice differentiable at all points of (a, b), and assume that f(0) = 0, f(1) = 1 and f(2) = 2. P
I need help to write this Proof. Please be as professional and as clear as possible in your response. Pay close attention to instructions. Consider the following subset B of R^3 (real Euclidean space): B={(x,y,z) in R^3 such that x=y+z} With the usual "component wise" addition and scalar multiplication in R^3, is B a
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).
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}.
Provide an in-depth introduction to help students understand the concept of a limit. Provide three detailed examples of finding limits.
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
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)
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?
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.
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.
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
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
Problem is in the attached file. Thank you.
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.
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
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
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.
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:
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
Why does it suffice to say that |S-S_n|< epsilon implies that S=sum(n=1 to infinity) a_n is uniformly convergent? I am looking for the reasoning behind it or some reading material on it.
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
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
1. Discuss the convergence and the uniform convergence of the series sum (f_n) where f_n(x) is given by a) x^n/(x^n+1) where x is greater than or equal to zero b) ((-1)^n))/(n+x) for x greater than or equal to zero.
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
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
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.
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)
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].
For each natural number n and each number x in (-1,1), define f_n (x)=√(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.
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