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Real Analysis

Real analysis : Measurable Functions

Q1. If 0<=a_1<=a_2<=a_3<=....,( 1,2,3 are the subscripts of a) 0<=b_1<=b_2<=b_3<=......(1,2,3 are the subscripts of b) and a_n --> a and b_n -->b Then prove that a_n*b_n -->a*b Q2.Let f: R --> R be monotonically increasing, i.e. f(x_1)<= f(x_2) for x_1< = x_2. Show that f is measurable. Hint: You may extend f to f':[-in

Radius of Convergence : Summations and Power Series

Please see the attached file for full problem description. a) Evaluate b) What is the radus of convergence .... .... To what simple function does this series converge? c) Is f(z)=... ... analytic near z= -1? and expand f(z)= ... in a power series near z= -1 can we predict the domain of convergence from the outset?

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.

Convergence

Please see the attached file for full problem description.

Limits

I attached a word document. Be sure to show me all of your work so that I can fully understand how to do the problems correctly. Thank you very much for your help.

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

Riemann Sum and Limit

Write out the Riemann Sum for R(f,P, 0, 2) for arbitrary n, f(x) = x2&#8722;3x+2, where each &#8710;xk = 2/n and ck = xk, simplify and use the formulas &#8721;n,k=1 k=(n(n+1))/2 and &#8721;n,k=1 k2=n((n + 1)(2n + 1))/6 to find the limit as n --> 1.

Limits

See attached lim (sin^2 3t)/2t t--->0

Limits

Please see the attached file for the fully formatted problem. Lim (1 - cos t)/2t t--> 0

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.

Summation Series

Summation Series. See attached file for full problem description.

Double Integral : Horizontal and Vertical Simple Methods

I have managed to evaluate the double integral using the horizontal simple method, and answer 63. But when I reverse the order (vertical simple method) I cannot reach the same answer of 63, I get 65. 1) Evaluate the double integral of f(x,y)=x+4y^2 over the triangular region with vertices (-2,2) (4,2) & (1,-1) Check that r

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 : Proof

I need a proof for "If f on [a,b] is continuous & 0 is not a member f([a,b]) then f is bounded away from 0."

Real Analysis : Proof using Summation Integrals

For numbers a1,....,an, define p(x) = a1x +a2x^2+....+anx^n for all x. Suppose that: (a1)/2 + (a2)/3 +....+ (an)/(n+1) = 0 Prove that there is some point x in the interval (0,1) such that p(x) = 0