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Real Analysis

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.

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

Radius of Convergence

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

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 - 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,

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.

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(

Real Analysis Problem

We have just finished up integration and are done with a first course in analysis, so chapters 1-6 of Rudin. We are also using the Ross and Morrey/Protter book. Please answer question fully and clearly explaining every step. Any solution short of perfect is useless to me. So if you are not 100% sure whether your answer is right,

Real Analysis Problem

We have learned Rolle, Lagrange, Fermat, Taylor Theorems in our Real Analysis class and we have finished differentiation. We just started integration. In this problem we are not supposed to use any material we haven't learned, ie integration. We are using the books by Rudin, Ross, Morrey/Protter. ****************************