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# Real Analysis

Real Analysis refers to the study of real valued functions. It particularly focuses on the properties of these functions, including but not limited to, the convergence and limits of sequences of real numbers, as well as the calculus of real numbers. In dealing with Real Analysis, it is important to understand the different terms. If a limit for a particular sequence exists, then the sequence is called convergent; however if the limit for a particular sequence does not exist, then the sequence is called divergent.

For example, if the sequence was 1, ½, 1/3 , ¼, 1/5, 1/6 and so on, this can be written in the following form:

(1/j)_j=1->inf

where,

j is the denominator of the function

inf is infinity

This sequence eventually converges to zero, because as j approaches infinity, the denominator gets extremely large, leaving the output of the function to be extremely small. However, the important aspect to consider is that the series never actually reaches zero, but it can get as infinitesimally close to it. This is why the study of limits in this particular context is considered under Real Analysis.

## BrainMass Categories within Real Analysis

### Sequences Limit Consequences

(a) Prove this operation: Let {xn} and {yn} be convergent sequences. The sequence{zn} where zn:=xn-yn converges and lim (xn-yn)=lim zn=lim xn-limyn What I attempted was this: Suppose {xn} and {yn} are convergent sequences and write zn:=xn-yn. Let x:=lim xn, y:=lim yn and z:=x-y Let epsilon>0 be given. Find M1 s.t. for

### Satisfying the properties of a metric

This problem from Methods of Real Analysis, 2nd edition, by Richard Goldberg. For P<x_1,y_1> and Q<x_2, y_2> define σ(P,Q)=| x_1- x_2 |+|y_1+y_2 | Show that σ is a metric for the set of ordered pairs of real numbers. Also, if: τ(P,Q)=max⁡(| x_1- x_2 |,|y_1+y_2 |) show that τ defines a metric for the same set

### Limits, Differentiation 8 Questions

1. Find the limit as x approaches infinity of the function sin 18x/10x 2. Find the limit as x approaches 6 of the function 3/(x-6)^2 3. Find the limit as y approaches 0 of the function (sin 11y)/13y or state that the limit does not exist. 4. Find the limit by substitution of the following: the limit as x approaches -2 o

### Series of Logical Arguments - Cutting Emissions

Your city has been warned by the EPA to cut sulfur dioxide and nitrogen oxide emissions or you will lose substantial federal funds. One citizen's group has advocated drastically reducing electricity use. A second group advocates reduced use of automobiles within city limits. Choose one or the other position to support. Develop a

### Series Calculation and Its Convergent Properity

(a) (i) Fine the sum of the integers from 17 to 92 inclusive. (ii) Hence find the value of: 92 ∑ (5+6i) i = 17 (b) For each of the sequences below, decide whether it converges, and if it does, state its limit. (i) a*subscript*n = 7 + 2(0.4)^n / 8(0.3)^n - 8 (n = 1, 2, 3, ... ) (ii) b *subscript* n

### Taylor series: open interval of convergence

The starting point for this problem is the representation of the natural exponential function by the corresponding Taylor series in powers of x: e^x= 1 + x +(1/2!)x^2 + (1/3!)x^3 + ......(1/n!)x^n + ........, x E R a) Let f(t) = e^(-t^2) , t E R. Define the Taylor series for f from the Taylor series for e^x. b) Let erf(x)=

### Taylor Series Calculus

The starting point for this example is the Taylor series for sine: sin (x) = x - (1/3!)x^3 + (1/5!)x^5 ?(1/7!)x^7 + ...... + (-1)^n * 1/(2n + 1)!*x^(2n+1) + ....... a) Let f(x) = { sin(x) / x if x doesn't equal 0 {1 if x doesn't equal 0 Show that f is infinitely differentiable on R (inclu

### Pointwise Limit and Uniformity

Let fk (x) = kxe^-kx, k = 1, 2, 3, . . . . a) Determine the pointwise limit f of the sequence {fk} infinity k=1 on [0, +infinity). b) Show that the sequence {fk} infinity k=1 does not converges to f uniformly to f on [0,+infinity). c) Show that the sequence {fk} infinity k=1 converges to f uniformly to f on [sigma,infinity),

### Pointwise limit question

Let fk (x) = (e^(-x^(2) /k) / k , k = 1, 2, 3, . . . . a) Determine the pointwise limit f of the sequence {fk} infinity k=1 on R. b) Show that the sequence {fk} infinity k=1 converges to f uniformly to f on R.

### Pointwise limit problem

Let fk (x) = x^(1/k), k = 1, 2, 3, . . .. a) Determine the pointwise limit f of the sequence {fk} infinity k=1 on the interval [0, 1]. b) Show that the sequence {fk} infinity k=1 does not converge to f uniformly to f on [0, 1]. c) Show that the sequence {fk} infinity k=1 converges to f uniformly to f on [sigma, 1], where 0 <

### Real Analysis: Convergent Subsequence of Limit Superior

Suppose the lim (subscript n --> infinity) sup a_n = theta is finite. Prove there exists a subsequence that converges to theta. Please view the attachment to see the full question, including necessary symbols.

### Subsequence that Converges to a Limit L

Let theta be a (finite) real number, and show that if (a_n) is a sequence such that the limit (as n goes to infinity) of sup a_n is theta, and if (a_n) has a subsequence that converges to a limit L, then L is less than or equal to theta. Please see the attached. Thanks

### Find e^A Using the Taylor Series.

HI Professor Shafrir, I have chosen you because I like your pedagogic method for explaining each steps. In the following attached problem, the professor gives us the solution but didn't explain the way. So I don't need the solution, I have it. All I need to understand is how to get from step 1 (the Taylor series) to step 2

### Values of x in a series

The following series is developed in calculus: ln (l + x) = x - (x^2/2) + (x^3/3) - (x^4/4) + . . . Over what range of values for x can this series be used to calculate ln(l + x)? (1) For all values of x (2) - l < x <= l (3) - l <= x <= l (4) - infinity < x <= l (5) |x| > l

### Radius of convergence of binomial series for fixed complex x

The generalized binomial coefficient for z in C (complex) and k = 0, 1, 2, ... is 1 if k = 0 and ( k ) ( z ) = [z(z-1) ... (z - k +1)] / k! if k >=1. For a fixed value of s in C we define the binomial series B_s*(z) = sum of n=0, ^infinity, ( s ) ( n ) z^

### Merton Medical Clinic: Project Analysis and NPV

The managers of Merton Medical Clinic are analyzing a proposed project. The project's most likely Net Present Value (NPV) is \$120,000, but, as evidenced by the following NPV distribution, there is considerable risk involved: Probability NPV 0.05 \$-700,000 0.2 \$-250,000 0.5 \$120,000 0.2 \$200,000 0.05 \$300,000 a. What a

### Evaluate Some Limits

Provide some guidance in approaching the following question: let c denote a complex constant then show that : lim (z^2 + c) = z0^2 +c z->z0.

### Introductory Real Analysis

Complete question 1, 4 and 8 only.

### Proof that uses limiting values of functions

Propose a definition for limit superior lim sup_x-->x_0;x belonging to E of f(x) and limit inferior lim inf_x-->x_0; x belonging to E of f(x) and then propose an analogue of the following for your definition and prove that analogue Let X be a subet of R, let f: X-->R be a function, let E be a subset of X, let x_0 be an adher

### Real analysis (Uniform continuous)

Let {fn} be a sequence of continuous functions, all defined on [a,b]. Suppose {fn(a)} is a diverging sequence of real numbers. Prove that {fn} does not converge uniformly on (a,b]. (Notice a is not included in this interval.) Assume {fn} converges unifromly and then use the diverging sequence of real numbers as a contradict

### Taylor and MacLaurin Series

Hi, Can you please assist with homework problems attached? Thanks.

### Real Analysis Question: Three Proofs

Let y be a positive real number. Choose x_1>sqrt(y) and let x_n+1=1/2(x_n + y/x_n), for all n>=1. 1) Prove {x_n} is monotonically decreasing and bounded. 2) Prove limit of x_n as n approaches infinity is sqrt(y) 3) Letting r_n=x_n-sqrt(y) show that r_n+1=r^2_n/2x_n < r^2_n/2sqrt(y) for all n>=1. Conclude that r_n+1<z(r_

### Real Analysis Sequences Questions

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### Proof Convergence and Limit Laws

Let (a_n)^infinity, n=m and (b_n)^infinity, n=m be convergent sequences of real numbers, and let x, y be the real numbers x : = lim_n-->infinity a_n and y : = lim_n-->infinity b_n Prove: a) The sequence (a_n b_n) ^ infinity, n=m converges to xy; in other words lim_n-->infinity (a_n b_n)= (lim_n-->infinity a_n) (lim_n-->infi

### Proof real exponentiation

Let x,y > 0 be positive reals, and let n, m >= 1 be positive integers, Prove: If x > 1, then x ^ 1/k is a decreasing function of k. If x < 1, then x ^ 1/k is an increasing function of k. If x=1, then x ^ 1/k = 1 for all k.

### Sales Tax on a Vehicle Purchase

1.) Sales Tax: Sales tax on the purchase of a vehicle is calculated on the net purchase price (which is the total purchase price minus the amount allowed by a dealer for any trade-in). The rate of tax for residents in Denver, CO is as follows: ?Colorado state tax: 2.90% ?Regional transportation district (RTD) tax: 1.20% ?D

### Power Series - Boundedness

** Please view the attachment for proper formatting of this problem ** (a) Suppose E and F are nonempty closed bounded subsets of C. Show that there exist z_0 E E and w_0 E F such that | z_0 - w_0 | = inf {| z - w | ; z E E, w E F}. (b) Show that this is not true if the boundedness condition on E and F is dropped. (

### Managerial Science Excel Sheet

The questions are in red on the Excel sheet. Monthly Demand (Number of New sign customer sign ups) Percentage of a time demand level occurs 1,000 5 2,000 15 3500 20 5,000 30 7,000 25 9,000 5 In addition to the historical demand, the comp

### Real Analysis: Isometrics

** Please see the attached file for the complete problem description ** Please show all steps involved. A function f from a metric space (X,d) onto a metric space (Y, (please see the attached file)) is called an isometry if: (please see the attached file) a) Show that an isometry is continuous, one-to-one, and its inve

### determine the order quantity that will result in minimum

Demand = 4,800 per year Order cost = 40 per order Holding cost = 25% of acquisition cost per unit per year Delivery size is equal to the order quantity. Range Cost per unit 1 - 399 10.00 400 - 1199 9.00 1200 - 4799 8.50 4800 an