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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 &#8721;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&#8804;1 See attached file for full problem description.

### Continuous, Real Valued, Linearly Dependent Functions and Matrix Determinants

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.

### Real Analysis : Subsequences and Convergence

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.

### Radius of Convergence of Complex Power Series

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).

### Convergence or Divergence of Series by the Ratio Test

I must use the ratio test to show that the following series converges: &#8734; &#931; (k^2 + i)/(k+i)^4 k=1

### Sequences and Limit Superior

Suppose that xn&#61664; 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

### Finding Limits of Functions

Find Limit. See attached for full problem description.

### 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

### Implicit differentiation. L'Hopital rule, Convergence and divergence of the series

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

### Evaluating a Trigonometric Limit

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

### Evaluating a Limit: Example

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

### Series : Uniform Convergence and No Zeros

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}

### 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

### Analysis of Limit Problem : Sequences of Positive Numbers

Please see the attached file for the fully formatted problem.

### Power Series Expansion, Radius of Convergence & Binomial Theorem

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

### Calculate the nth term for given series

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

### 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.

### Complex Sequences and Series: Convergence or Divergence?

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

### 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

### Convergence or Divergence of Infinite Series : Integral Test

Verify that the infinite series diverges. &#8734; &#8721; n/(2n+3) n=1

### Convergence or Divergence of Infinite Series

Use the Integral Test to determine the convergence or divergence of the series. &#8734; &#8721; ln n/ n^3 n=2

### Convrergence or Divergence of Infinte Series

Use the Integral Test to determine the convergence or divergence of the series. &#8734; &#8721; ne^(-n/2) n=1

### 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

### Convergence or Divergence of Series

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