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Real Analysis

Real Analysis: Absolutely Continuous

See attached file for full problem description. Problem 4 Only. If f:[a,b]-->R is absolutely continuous then |f(e)| = 0 for all E⊂ [a,b] with |E| = 0.

Power Series and Radius of Convergence

Consider the power series ∑anxn for which each coefficient an is an integer. Prove that this series has a radius of convergence, R, where either R=positive infinity or R≤1 See attached file for full problem description.

Sequences and Limit Superior

Suppose that xn x and the sequence (yn) is bounded. Show that ___ ___ lim (xn + yn) = lim xn + lim (yn). ___ I know that since (xn) converges lim xn = lim (xn) and that ___ __ ___

Taylor Series Problem

Write the Taylor series about x=0 for ([cos(X^3)]/x) and ([cos(7X)]/x^3) I would like to see the step by step process for how this works. I know the standard series for any variable of cos(x), but would like to understand how dividing by x changes the problem keywords : find, finding, calculating, calculate, determine,

Limits

Please explain how/why: lim x---> - 4 x^2 - x - 20 / x+4

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

Finding Limits and Continuity

A) Find lim x---> 2+ C(x) b) Find lim---> 2- C (x) c) Find lim x---> 2 C (x) d) Find C (2) e) Is C continuous at x=2? At 1.95? See attached file for full problem description.

Limits : L'Hopital's Rule

Please see the attached file for the fully formatted problems. keywords : find, finding, calculating, calculate, determine, determining, verify, verifying, evaluate, evaluating, calculate, calculating, prove, proving, L'Hospital's, L'Hospital

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

Taylor Series Expansions, Residues and Larent Series

a) Give the infinite Taylor series expansions for the three functions e^z, sin z, cos z. b) Write 5 nonzero terms, including the one that determines the residue, for the Laurent series of e^z / z^4 centered at 0.

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