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# Real Analysis

### Real Analysis : Properties of Sets and Relations

Please see the attached file for the fully formatted problems.

### Singularities, Convergence of Taylor Series, Residues and Contour Integral

Consider the complex function: f(z) = 1/(z^2 + z + 1)(z + 5i) a) Find and classify the singularities of f(z); b) Without finding the series explicitly, determine the region of uniform convergence of the Taylor series taken about the origin; c) Find the residues of f(z) at each of the singular points; d) Find the val

### Integrals and Convergence Evaluated

Determine whether the integral dx / x^2 which has a an upper limit of 3 and lower limit of -2 converges or diverges. Evaluate the integral if it converges.

### Quality Control : Control Limits and p-Charts

Hunter Nut Company produces cans of mixed nuts, advertised as containing no more than 20% peanuts. Hunter Nut Company wants to establish control limits for their process to ensure meeting this requirement. They have taken 30 samples of 144 cans of nuts from their production process at periodic intervals, inspected each can, and

### Evaluate the Limit

Evaluate lim x => 16 (sqrt(x-4) / x-16)

### Find the Maclaurin series for f(x) using the definition of a Maclaurin series and find the radius of convergence.

Please see the attached file for the fully formatted problems.

### The Sum of Converging Series

Consider the following series. SUM (n =0, infinity) of (x + 7)^n / 4^n a) Find the values of x for which the series converges. (Enter the smaller number first.) b) Find the sum of the series for those values of x.

### The Sum of Converging Series

Find the sum of the series, if it converges. Otherwise, enter DNE. SUM (n = 1, infinity) of (-5)^(n-1) / 8^n

### Sequence converges or diverges

1. Determine whether the sequence converges or diverges. If it converges, find the limit. If it diverges write NONE.

### Sequence Convergence and Limit Proof

Suppose {p_n} converges to p. Prove that there is at most one alpha for which the limit as n goes to infinity of |p_n+1-p|/(|p_n-p|^alpha) is a positive finite number. (See attachment for mathematical notation)

### Explicit Iterative Functions : Sequences and Order of Convergence

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### Multiplicity of a Root and Taylor's Theorem

Let f(x) be defined as f(x) = tanx/x if x =/ 0 = 1 if x = 0 and let g(x) = f(x) - 1. Then g(x) is continuous at x = 0 and, in fact, g(x) has derivatives of all orders at x = 0. Determine the multiplicity of the root g(x) has at x = 0. Hint: Apply Taylor's Theorem.

### Real Analysis Proofs : Set Operations

Provide counterexamples for each of the following.... From Set Theory, Set Operation. College level proof before real analysis. Please give formal proof. Please explain each step of your solution. Thank you.

### Real Analysis Proofs : Set Operations

From Set Theory, Set Operation College level proof before real analysis. Please give formal proof. Please explain each step of your solution. Thank you.

### Provide either a proof or a counter example for each....

College level proof before real analysis. Please give formal proof. Please explain each step of your solution. Thank you.

### Use the technique of working backward from the desired conclusion to prove that...

College level proof before real analysis. Please give formal proof. Please explain each step of your solution. Thank you.

### Real Analysis Integer Proofs

College level proof before real analysis. Please give formal proof. Please explain each step of your solution. Thank you.

### Real Analysis : Real Numbers

College level proof before real analysis. Please give formal proof. Please explain each step of your solution. If you have any suggestion or question to me, please let me know. Thank you. Please see attached file for full problem description. I. Three real numbers and have the property that . Prove that at least

### Example Functions and Limits Problems

1. Express the distance between the point (3, 0) and the point P (x, y) of the parabola y = x2 as a function of x. 2. Find a function f(x) = xk and a function g such that f(g(x)) = h(x) = 3x + x2 3. Find the trigonometric limit: lim x-tan 2x/sin 2 x → 0

### Convergence and Interchanging limits

Suppose that f_k -> f uniformly on (0,1). Let 0 < x < 1. Suppose that lim f_k(t) = A_k for k=1,2,... Show that {A_k} converges and lim f(t) = LIM A_k. That is show lim LIM f_k(t) = LIM lim f_k(t). Where lim represents the limit as t approaches x and LIM represents the limit as k approaches infinity.

### Piecewise Functions, Derivatives and Limits

1. A piecewise function is given. Use the function to find the indicated limits, or state that a limit does not exist. (a) lim is over x gd - f(x), (b) lim is over x gd + f(x), and (c) lim is over xgd f(x) f(x) = { x^2 - 5 if x < 0 } { -2 if x >= 0 } : d = -3 (a) -5 (b) -2 (c) does not exist

### Derivatives, Integrals, Limits and Convergence

Please see the attached file for the fully formatted problems. Question 1 Differentiate the function f(x) = (a) xlnx - x (b) x5lnx (c) (lnx)2 (d) 1-x ________________________________________lnx Question 2 Figure 2.1 ?(x) = ln ^/¯ (9-x2) ________________________________________(4+x2)

### There are twelve problems involving functions, continuity, finding slope using predictor formula, tangent line to a curve, trajectory of a projectile, finding limits, finding limits using squeeze law and continuity of functions.

Please refer to the attached file to view the complete questions. ======================================== Question 1 Figure 1.1 y = f(x) = (2x+4)2 - (2x - 4)2 . Apply the slope predictor formula to find the slope of the line tangent to Figure 1.1. Then write the equation of the line tangent to the graph of f at

### Real Analysis: Integrable Functions

Let f:[0,1]-->R be a Riemann integrable function. Prove that lim n-->∞ ∫ (from 0 to 1) x^n f(x)dx = 0. I do not know where to begin on this problem. It seems like it should be easy though.

### Real Analysis : Differentiable Functions

Does there exist a differentiable function f: R-->R such that f'(0) < 0 for all x &#8804; 0 and f'(x) > 0 for all x > 0? Give an example of such function or prove that it does not exist.

### Real Analysis : Continuous Functions

Let f:[-1,1]-->R be a continuous function such that f(-1)=f(1). Prove that there exists x &#1028; [0,1] such that f(x)=f(x-1).

### Maclaurin Series, Lagrange Interpolation Polynomial and Newton's Interpolation Formulas

1.Expand the following function into Maclaurin Series (see attached file) using properties of the power series. 2. The Lagrange interpolation polynomial may be compactly written as is a shape function. Sketch the shape function in a graphic form. 3. Write a forward and backward difference Newton's interpolation formulas b

### 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

### Real Analysis: Riemann Stieltjes Integral

Please see the attached file for the fully formatted problems.