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# Real Analysis

### Series Convergence and Divergence

Please see attached file for full problem description. 1) Consider the series where . Show that and for . 2) Use the result of the previous problem to find . 3) The series converges. Find its sum. 4) Determine whether the series converges or diverges. Fully justify your answer. 5) Determine wheth

### Trigonometric Limits

Use one-sided limits to find the limit or determine that the limit does not exist. 16-x^2 /4-x lim x => 4 Find the trigonometric limit: sin3x/2x limx => 0 Please show work.

### Measure Theory and Dominated Convergence Theorem

Please see the attached file for the fully formatted problems. I have provided a solution to the attached problem. I do not understand or like the solution - I was hoping you could provide an alternate solution or expand upon the solution I have provided in more detail. Exercise (moment-generating function). ? Let X be

### Function of Limits

The function has a limit as f(x) = (1/x) + 3 has a limit of L=3 as x approaches x. This means that if x is sufficiently large (that is if x > N for some number N), the values of f(x) are closer to L=3 than a number epsilon > 0. a) Sketch the graph y=(1/x) +3 and a horizontal strip of points (x,y) such that (if y

### Analysis: Finding a Limit

Find limit x>> -1^+ f(x) f(x)= x - 2, for x <= 3; x - 1, for x > 3 Also: When you first begin to draw the graph of f(x), why do you start where you do? How high do I go when drawing a graph? How do I know when to stop?

### Limit

Evaluate the limit: lim x cot x x&#8594;0

These problems may or may not need PHSTAT again. All is the same as priors posting the responses in a MS Word document. Attached are: -Problems [MS Word] -Spreadsheet [Excel]

### Some problems on network models

I have attached some quantitative Analysis Questions that I need help solving so I can add to my study guide. The book used was Quantitative analysis for Management by Barry Render but as you already know these concepts are found in any Quantitative Analysis book.

### Decision Analysis : Influence Diagrams, Decision Trees and Optimistic, Conservative, and Minimax Regret Approaches

3. Southland Corporation's decision to produce a new line of recreational products resulted in the need to construct either a small plant or a large plant. The selection of plant size depends on how the marketplace reacts to the new product line. To conduct an analysis, marketing has decided to view the possible long-run dema

### Remainder Estimation Theorem and Euler's Formula

1.) Given that 1-x+x^2+...+(-x)^n is a power series representation for 1/(1+x), find a power series representation for (x^3)/(1+x^2). 2.) The Maclaurin series for f(x) is: 1+2x+((3x^2)/(2))+((4x^3)/(6))+...+(((n+1)x^n)/(n!))+... Let h(x)= (the integral from 0 to x) f(t) dt. Write the Maclaurin series for h(x).

### Taylor Series and Systems of Vector Equations

1 Write the Taylor series with center zero for the function f(x) = ln(l+x^2). 2 Given a =2i +3j, b = 3i +5j , and c = 8i + 11j express c in the form ra+ sb where r and s are scalars.

### Limits: True or False

Please help with the following problem. Provide step by step calculations. True or false: a)lim x--->2- f(x) = 3 b)limx--->2+ f(x) =0 c)lim x--->2- f(x) = lim x--->2+ f(x) d) lim x--->2 f(x) exists e) lim x--->4 f(x) exists f) lim x--->4 f(x) = f(4) g) f is continuous at x=4 h) f is continuous at x=0 i) lim x--->3

### Implicit differentiation. L'Hopital rule, Convergence and divergence of the series

1. Express the distance between the point (3, 0) and the point P (x, y) of the parabola y = as a function of x. 2. Find a function f(x) = and a function g such that f(g(x)) = h(x) = 3. Find the trigonometric limit: . 4. Given , use the four step process to find a slope-predictor function m(x). Then write an eq

### Real Analysis : Norms and Bounded Sets

8. Fix an n-dimensional real vector space V with n a positive integer greater than 1. If you want to take V to be R, fine. Consider non-empty open sets B C V with the following properties: (a) B is bounded and convex (contains the line segment through any two of its points); (b) If VEB,then there is a number t0>0 for which tv

### Computing U, H, F, G, S, and mu for nitrogen gas.

For a mole of nitrogen (N_2) gas at room temperature and atmospheric pressure, compute the internal energy, the enthalpy, the Helmholtz free energy, the Gibbs free energy, the entropy, and the chemical potential. The rotational constant epsilon for N_2 is 0.00025 eV. The electronic ground state is not degenerate.

### Finding a Limit using Riemann Sums

Evaluate (lim)(sin(Pi/(n))+sin((2*Pi)/(n))+sin((3*Pi)/(n))+***+sin((n*Pi)/(n)))/(n) by interpreting it as the limit of Riemann sums for a continuous function f defined on [0,1]. keywords: integration, integrates, integrals, integrating, double, triple, multiple

### A piece of electronic equipment used for aviation has three elements connected in series, or sequence

3. A piece of electronic equipment used for aviation has three elements connected in series, or sequence. The reliability of each of the three elements is as follows: Element A: 0.92 Element B: 0.94 Element C: 0.91 a) Draw how these three elements are connected. b) What is the r

### Convergence of Power Series

1 Determine whether the series converges absolutely, converges conditionally, or diverges. &#8734; &#931; 2&#8729;4&#8729;6&#8729;&#8729;(2n)/2&#8319;(n+2)! n=1 2 Calculate sin 87° accurate to five decimal places using Taylor's formula for an appropriate functio

### Power Series Proof

Define the set R[[X]] of formal power series in the indeterminate X with coefficients from R to be all formal infinite sums sum(a_nX^n)=a_0 +a_1X+a_2X^2+... Define addition and multiplication of power series in the same way as for power series with real or complex coeficients,i.e extend polynomial addition and multiplication t

### General Vector Taylor Series Expansion: Measure of deviation

2. Arfken, p. 342, 5th Ed. (p. 359, 6th Ed.), ) Prob. 5.6.7. Use the General Vector Taylor Series Expansion For a General Function, cI) (r) = (I) (x, y, z) , Of a Three-Dimensional Vector Coordinate, Expressed In Cartesian Coordinates, which is Expanded About the Origin, r = 0 Or x = y = z = 0 , Where 0(0 = (1)(x', y', z') 1 ir,

### Taylor Series Expansion and derivation of the Euler Formula

Show that (a) sin x = summation (0-infinity) (-1)^n x^(2n+1)/(2n+1)! (b) cos x = summation (0-infinity) (-1)^n x^(2n)/(2n)! Use the Taylor series expansion around the origin, f(x) = summation (0-infinity)[x^n/n!]f^n(0), and derive the power series expansions for sin x , cos x and e^x. Then write out the first few real

### Real Analysis : Limits and Continuity of Piecewise Functions

Determine whether or not each of the following limits exists. Discuss also the continuity of each of the following functions at given point c. Give reasons to your answers. Please see the attached file for the fully formatted problems.

### 3-Sigma Control Limits

Jim Outfitters makes custom fancy shirts for cowboys. The shirts could be flawed in various ways, including flaws in the weave or color of the fabric, loose buttons or decorations, wrong dimensions, and uneven stitches. Jim randomly examined 10 shirts, with the following results: Shirt Defects 1 8 2 0 3 7 4 12 5 5 6 10

### Finding Area Using Sums and Limits

Given f(x) = x^2 + 3, find the exact area A of the region under f(x) on the interval [1, 3] by first computing n &#931;f(xi)&#916;x and then taking the limit as n-->&#8734;. i=1 Please see the attached file for the fully formatted problems.

### Problem set For the equation ?(x)= x^(1/2) a) Find the Taylor polynomial of degree 4 of at c = 4 b) Determine the accuracy of the polynomial at x = 2. Question (2) Find the Maclaurin series in closed form of Question (3) Use the chain rule to find dw / dt, where w = x^2 + y^2 + z^2, x=(e^t) cos t, y=(e^t) sin t, z=(e^t), t=0 Question (4): Find the critical points and test for relative extrema: ?(x,y)=2(x^2)+2xy+(y^2)+2x-3

1. For the equation ?(x)= x^(1/2) a) Find the Taylor polynomial of degree 4 of at c = 4 b) Determine the accuracy of the polynomial at x = 2. 2. Find the Maclaurin series in closed form of a) ?(x)=((1) / ((x+1)^2) b) ?(x)=ln ((x^2)+1) 3. Use the chain rule to find dw / dt, where w = x^2 + y^2 + z^2, x=(e^t) cos t, y=(

### Real analysis: Lebesgue Integral

Prove theorem 7.3 in notes attached. Section 7: The Lebesgue Integral Definition 7.1 Let L be the set of real-valued functions f such that for some g and h in f=g-h almost everywhere. The set L is called the set of Lebesgue integrable function on and the Lebesgue integral of f is defined as follows: . Theorem 7

### Real Analysis : Equicontinuous

If E is equicontinuous in C(X,R), show that E-bar (the closure of E) is also equicontinuous. keywords: equicontinuity

### Testing Series for Convergence

Test for convergence or divergence 1.) sum from n=1 to infinity of (e^1/n)/(n^2) 2.) sum from j=1 to infinity of (-1)^j * ((sqrt j)/(j+5)) 3.) sum from n=2 to infinity of (1/((1+n)^(ln n)) keywords: tests

### Sequences and Series (20 Problems): Partial Sums, Convergence and Divergence

Please do all problems below step by step showing me everything. Do simply as possible so I can clearly understand without rework. Adult here relearning so show all work, etc. OK, some said cannot read problems, but do not have a scanner with me know, so typed them in below. Sorry for any problems, but this shopuld clear up

### Limit Proofs

Prove the following a) If lim n-->infinity (a_n*b_n) exists and lim n--> infinity (a_n) exists, then lim n -->infinity (b_n) exists. b) If lim n--> infinity (a_n) = 0 and {b_n} is bounded, then lim n-->infinity (a_n*b_n) exists and equals 0. c) If lim superior (a_n) exists, then {a_n}_n is bounded above.