### 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.

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.

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.

Let f1,...,fk be continuous real valued functions on the interval [a,b]. Show that the set {f1,...,fk}is linearly dependent on [a,b] iff the k x k matrix with entries b <fi,fj> = ∫ fi(x)fj(x)dx has determinant zero. a See attached file for full problem description.

Let fn(x) = cos(nx) on R. Prove that there is no subsequence fnk converging uniformly in R. Please see the attached file for the fully formatted problems.

I have to find the radius of convergence for the following series: Sum from j = 0 to infinity of z^(3j)/2^j, and the answer according to my book is R = 2^(1/3).

I must use the ratio test to show that the following series converges: ∞ Σ (k^2 + i)/(k+i)^4 k=1

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 ___ __ ___

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,

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

Find Limit. See attached for full problem description.

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

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.

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

1. Express the distance between the point (3, 0) and the point P (x, y) of the parabola y = as a function of x. 2. Find a function f(x) = and a function g such that f(g(x)) = h(x) = 3. Find the trigonometric limit: . 4. Given , use the four step process to find a slope-predictor function m(x). Then write an eq

Please solve for the following: Evaluate the limit of (sin[x]-x)/(x^3) as x approaches 0.

Evaluate the limit: (x^2)/(ln[x]) as x approaches positive infinity.

Fix R>0. Show that, if n is large enough, then P_n(z)=1+z+z^2/2!+z^3/3!+...+z^n/n! has no zeros in {z:|z|<=R}

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

Please see the attached file for the fully formatted problem.

Please see the attached files for the fully formatted problems. This problem is from Introduction to Analysis (Maxwell Rosenlicht).

What is the nth term for the following: 1, 1/2, 3, 1/4, 5, 1/6.

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.

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.

Please see the attached file for the fully formatted problems. Make sure to show all work when solving.

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

Verify that the infinite series diverges. ∞ ∑ n/(2n+3) n=1

Use the Integral Test to determine the convergence or divergence of the series. ∞ ∑ ln n/ n^3 n=2

Use the Integral Test to determine the convergence or divergence of the series. ∞ ∑ ne^(-n/2) n=1

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

Determine the convergence or divergence of the series. See attached file for full problem description. ∞ Σ (n+1)/(2n-1) n=1