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Real Analysis

Real Analysis

Please see the attached file for the fully formatted problems. Suppose that is not a perfect nth power, i.e K is not equal to (a) Prove that is not a member of Q, the set of all rational numbers. (b) Infer that the nth root of a natural number is either a natural number or it is irrational.

Finding antiderivatives

Antiderivative of: (t5 + 6t3) dt

Uniform Convergence of Series of Functions

Show that the sum from 0 to infinity of (1-x)x^n does not converge uniformly on [0,1]. What subintervals of [0,1] does it converge uniforlmly on?

Fifth Term of the Series

What is the fifth term of the series an = n + an-1' if a0 = -3.

Real-Life Application : Examples of Data Modeled Using a Linear Formula

Find an article through newspapers, magazines, professional journals, etc and find at least two examples of data that are best modeled using linear formulae. Describe the importance of each example and why a linear model is appropriate for the data. Note that we are referring to a linear model not simply a time chart where dots

Limit of trigonometrical functions

Evaluate Limit[tan (a*theta)/sin(b*theta)] as theta approaches zero.

Calculating an infinite limit

Calculate limit (1/x^2) as x approaches infinity

Proving a limit exists

Prove that Lim (3x)=6 as x approaches 2 using Epsilon-Delta definition.

Convergence of Series

Determine whether the series Sum(n!/n^n) n=1..infinity is absolutely convergent, conditionally convergent or divergent.

Series Test

Test the series (in the attached file) for convergence or divergence by using the Comparison Test or the Limit Comparison Test.

Real Analysis : Finding a Maximum using Lagrange Multipliers

Please see the attached file for the fully formatted problem. What is the maximum of F = x1 +x2 +x3 +x4 on the intersection of x21 +x22 +x23 + x24 = 1 and x31+ x32+ x33+ x34= 0? As this is an analysis question, please be sure to be rigorous and as detailed as possible.

Verify Fubini's Theorem for an integral evaluated over an equilateral triangle.

Verify Fubini's Theorem for an integral evaluated over an equilateral triangle. (My notes from class-make a function up, similar to something used in question 1, but change powers of x & y.) You should discuss fully the reasons for the limits of integration in your solution.

Real Analysis : Mean Value Theorem

Let f(x) be integrable on [a,b], and let g(x) be nondecreasing and continuously differentiable on [a,b]. Let {p be element of P} be a partition of [a,b], and define U(f,g,p) = SIGMA (Mi(g(the ith term of x) - g(the (i-1)th term of x))) as i=1 to n L(f,g,p) = SIGMA (Ni(g(the ith term of x)-g(the (i-1)th term of x))) as i=1 t

Real Analysis :Problem Prove a function is integrable over [a,b]

Let f:[a,b] mapped to the Reals be a function that is integrable over [a,b], and let g:[a,b] mapped to the Reals be a function that agrees with f except at two points. Prove g is integrable over [a,b].

The problem is to determine the radius of convergence of the Taylor Series for each of the functions below centered at x. We are to explain our conclusion in each case. I would like to see how to work each problem (including what the Taylor Series is) and what the explanation is. a) centered at and NOTE: I know the

Infinite Series of Real Numbers (Cesaro summable)

Infinite Series of Real Numbers (Cesaro summable)

Power and Taylor series

Interval of Convergence of a power series a. Consider the Power series sum of series from n=1 to infinity of FnX^n. Use the ratio test to determine the open interval on which the pwr series converges. b. Show that the Taylor series of the Fcn f(x) = x/(1-x-x^2) about x=0 is given by: x/(1-x-x^2) = sum of series at

Real Analysis : Young's Inequality

Note: * = infinite Suppose that the function f:[0,*)->R is continuous and strictly increasing, with f(0) = 0 and f([0,*)) = [0,*). Then define F(x) = the integral from 0 to x of f and G(x) = the integral from 0 to x of f^-1 for all x>=0 (a) Prove Young's Inequality: ab <= F(a) + G(b) for all a >= 0 and b >= 0 (b) N

Real Analysis: Geometric Interpretation in Terms of Areas

Note: * = infinite Suppose that the function f:[0,*) -> R is continuous and strictly increasing, and that f:(0,*) -> R is differentiable. Moreover, assume f(0) = 0. Consider the formula: the integral from 0 to x of f + the integral from 0 to f(x) of f^-1 =xf(x) for all x>= 0. How can I provide a geometric interpretation

Real Analysis: Criteria for Integrability

Suppose the continuous function f:[a,b]->R has the property that: The integral from c to d f<=0 whenever a<=c<d<=b Prove that f(x)<=0 for all x in [a,b]. Is this true if we require only integrability of the function?

Real Analysis of Criteria for Integrability

Suppose that the function f:[a,b]->R is integrable and there is a postive number m such that f(x) >= m for all x in [a,b]. Show that the reciprocal function 1/f:[a,b]->R is integrable by proving that for each partition P of the interval [a,b], U(1/f,P) - L(1/f,P) <= 1/m^2[U(f,P) - L(f,P)]

Real Analysis - Riemann Integrability

Please see the attached file for the fully formatted problems. Prove that if f is integrable on [0, 1], then lim n !1 Z 1 0 x n f(x)dx = 0 Since this problem is an analysis problem, please be sure to be rigorous. It falls under the chapter on Integrability on R , where they define partition, refinement of a partition,

Real Analysis: Riemann Integrability

Please see the attached file for the fully formatted problem. Let E = { 1/n : n 2 N } . Prove that the function f(x) = ? 1 x 2 E 0 otherwise is integrable on [0,1]. What is the value of R 1 0 f(x)dx? Since this problem is an analysis problem, please be sure to be rigorous. It falls under the chapter on Integrabili

MacLaurin Series And Laplace Transforms : Absolute Convergence

Find MacLaurin Series for the given function f. Use the linearity of the Laplace Transform to obtain a series representation L(f)=F(s) Determine 5 values for which the series converges absolutley (and uniformly). Also show the Laplace transform exists, i.e. that it has exponential order alpha. Here are the functions. A) f

Real Analysis : Proof of a Constant

Please see the attached file for the fully formatted problem. Let > 0. Prove that log x  x for x large. Prove that there exists a constant C such that log x  C x for all x 2 [1, 1 ), C ! 1 as ! 0+, and C ! 0 as ! 1 Please justify all steps and be rigorous because it is an analysis problem. (Note: The probl

Real Analysis: Mean Value Theorem

Let r e-1/x2 i(x) = i 0 x740 x = 0 Show that the nth derivative of 1(x) exists for all n E N. Please justify all steps and be rigorous because it is an analysis problem. (Note: The problem falls under the chapter on Differentiability on IR in the section entitled The Mean Value Theorem.)

Composition Series

Write a composition series for the rotation group of the cube and show that it is indeed a composition series.

Evaluating Limits: Epsilon-Delta Definition and Limit Theorems

Evaluate the following limits using the epsilon - delta definition and the limit theorems. a) lim {x -> 0} sin x sin (1/x^2) b) lim {x -> Infinity} (x^3 + 1)/(x^3 cos(1/x) + x^2 - 1) Please also show how you came up with the answer.

Functions: Limit Definitions

Using the definition of a limit (rather than the limit theorems) prove that lim {x -> a+} f(x) exists and find the limit in each of the following cases a) f(x) = x/|x|, a = 0. b) f(x) = x + |x|, a = -1. c) f(x) = (x - 1)/(x^2 - 1), a = 1. In which cases do lim {x -> a-} f(x) and lim {x -> a} f(

Calculating the limits of a sequence.

Use two methods to calculate the limit of the following sequence. x(n)=1/n^3+2^2/n^3+...+n^2/n^3