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Real Analysis

Sums of Convergent Series

How many terms of the convergent series do you need in order for the partial sum Sn to estimate the actual sum of the series correctly to four decimal places? ( The sum of the series from n=1 to infinity of (-1) ^5n divided by Ln n.)

Some problems on network models

I have attached some quantitative Analysis Questions that I need help solving so I can add to my study guide. The book used was Quantitative analysis for Management by Barry Render but as you already know these concepts are found in any Quantitative Analysis book.

Terms of Power Series

4. Let f be the function by f(x)e^(-2x^2) a. Find the fist four nonzero terms and the general term of the power series for f(x) about x = 0. b. Find the interval of convergence of the power series for f(x) about x =0. Show the analysis that leads to our conclusion. c. .... Please see the attached file for the fully formatted

Convergence and Divergence of Series

Page Number: - 568 1 / (b, e, g, h) Page Number :- 576 1 / (c, d), 5 / ( a, c, d, f) Page Number: - 579 1 /( e , f, h), 3 Please explain all steps.

Remainder Estimation Theorem and Euler's Formula

1.) Given that 1-x+x^2+...+(-x)^n is a power series representation for 1/(1+x), find a power series representation for (x^3)/(1+x^2). 2.) The Maclaurin series for f(x) is: 1+2x+((3x^2)/(2))+((4x^3)/(6))+...+(((n+1)x^n)/(n!))+... Let h(x)= (the integral from 0 to x) f(t) dt. Write the Maclaurin series for h(x).

Taylor Series and Systems of Vector Equations

1 Write the Taylor series with center zero for the function f(x) = ln(l+x^2). 2 Given a =2i +3j, b = 3i +5j , and c = 8i + 11j express c in the form ra+ sb where r and s are scalars.

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

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

Find Limit

Find Limit. See attached for full problem description.

Limits: True or False

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 f(x) = lim x--->5 f(x) j) f is continuous at x = 2 See attached file for d

Find limits

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

Limits

Evaluate the limit: (x^2)/(ln[x]) as x approaches positive infinity. keywords: finding, find, calculate, calculating, determine, determining, verify, verifying, evaluate, evaluating

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