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# Real Analysis

### Real Analysis: Set Measures and Measurability

9. Let F be a closed subset in R, and ... the distance from x to F, that is, ....... Clearly, ....) whenever .... Prove the more refined estimate .... for a.e. xEF, that is,..... [Hint: Assume that x is a point of density of F.] Please see the attached file for the fully formatted problems.

### 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

### Find the radius of convergence and interval of convergence of power series.

1. Find the radius of convergence and interval of convergence of series. 2. A function f is defined by f(x) = 1+ 2x + x^2 + 2x^3 +x^4+...... that is, its coefficients are =1 and =2 for all n> =0. Find the interval of convergence of the series and find an explicit formula for f(x). 3. Suppose the radius of convergen

### 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.

### Semi-Continuous Function on a Compact Subset

Show that if f is an upper semi-continuous function on a compact subset K of R^p with values in R, then f is bounded above and attains its supremum on K. Edit: R^p--Let p be a natural number and let R^p denote the collection of all ordered "p-tuples"---i.e. (x_1, x_2,..., x_p) with x_i being a real number

### 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

### Proof: Sequences and Limits Example Problem

If {sn}∞ n=1 is a sequence of real numbers such that sn ≤ M for all n and lim n--> ∞ sn =L; prove that L ≤ M. Is the statement true if we replace both inequalities with "<"? See attachment for full equation.

### 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

### Real analysis - Proving Closure of a Ball

Let (X,d) be a metric space. Define a closed ball with center x and radius r to be the set B[x;r]={y:d(x,y)<=r}. Prove that B[x,r] is a closed set.

### Real analysis - open and closed sets

(See attached file for full problem description) 1. In the metric space show that: a. Any open interval of the form (a,b), (a, ), or (- ,b) is an open set. b. A close interval [a,b] is a closed set. c. Any interval of the form [a, ) is a closed set.

### 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.

### Sequences and Limits

Consider the real sequence {x_n}_n generated by the iteration scheme x_n+1 = x_n(2-ax_n), for n = 0, 1, 2, ...... where a>0 and x_0 satisfying 0 < x_0 </= 1/a. a. Prove 1/a>/=x_n>0 for all n. b. Prove x_n>/=x_n-1. c. Conclude that lim n-->infinity

### Arithmetic Series : Finding d, the difference between any 2 terms.

Use the arithmetic series of numbers 1, 3, 5, 7, 9,...to find the following: What is d, the difference between any 2 terms? Please show me how you get the answers you got and any helpful resources. Using the formula for the Nth term of an arithmetic series, what is 101st term? Using the formula for the sum of an arithme

### Dense Subset, Continuity and Uniform Convergence

Let E C R1 and let D be a dense subset of E. If are continuous real-valued functions on E for n=1,2,..., and fn converges uniformly on D, prove that fn converges uniformly on E. (See attached file for full problem description) I am using the book Methods of Real analysis by Richard Goldberg.

### Prove that the Series of Functions Converges Uniformly

(See attached file for full problem description with equations) --- 9.3-5 Let {f_n} (from n - 1 to infinity) be a sequence of functions on [a,b] such that (f_n)'(x) exists for every x is an element of {a,b](n is an element of I) and (1) {(f_n)(x_0)} (from n=1 to infinity) converges for some x_0 is an element of [a,b]. (2

### Sequences and Uniform Convergence

Let {fn} infinity-->n-1 be a sequence of continuous real-valued functions that converges uniformly on the closed bounded interval [a, b]. For each n&#1028; I let Fn(x) = &#8747; x--> a fn(t)dt a<x<b Show that {fn} infinity-->n-1 converges uniformly on [a,b]. (Hint: Use 9.2F) Theorem 9.2F;