Real Analysis Problem

Show that there is a countable set F of functions of form x--->a_0 + a_1cosx + a_2cos(2x) + ... +a_ncos(nx) such that every continuous real-valued function on [0,pi] is the uniform limit of a sequence of functions (f_n) in F.

Real Analysis - Applications of Mean Value Theorem

Show that if the roots of the polynomial p are all real, then the roots of p' are all real. If, in addition, the roots of p are all simple, then the roots of p' are all simple.

Discuss the differentiability of each of the following functions at all real numbers and find its derivative at those real numbers at which it is differentiable.

Discuss the differentiability of each of the following functions at all real numbers and find its derivative at those real numbers at which it is differentiable. See attached file for full problem description.

Convergence or Divergence of Integrals

The problems in the file submitted are from an undergraduate course in real Analysis. If you are able to work the problems, please detail any theorems or lemmas used in your solutions. The book we are using is titled "The Elements of Real Analysis" by Robert G. Bartle. We are working on derivatives and integrals, but have not ...continues

Integration Proof

The problem in the file submitted is from an undergraduate course in Real Analysis. If you are able to work the problems, please detail any theorems or lemmas used in your solutions. The book we are using is titled "The Elements of Real Analysis" by Robert G. Bartle. We are working on derivatives and integrals, but have not s ...continues

Evaluate Integrals

The problems in the file submitted are from an undergraduate course in real Analysis. If you are able to work the problems, please detail any theorems or lemmas used in your solutions. The book we are using is titled "The Elements of Real Analysis" by Robert G. Bartle. We are working on derivatives and integrals, but have not ...continues

Series Problem

Problem: Show that the convergence of a series is not affected by changing a finite number of its terms. (of course, the sum may well be changed).

Series Proof

Problem: Show that if a convergent series of real numbers contains only a finite number of negative terms, then it is absolutely convergent.

I have a function that is differentiable on [a,b] and I am trying to figure out which scenario is more restrictive on the function

I have a function that is differentiable on [a,b] and I am trying to figure out which scenario is more restrictive: a) the function is a Lipschitz function with a Lipschitz constant L in (0,1) or b) the absolute value of f'(x) is less than one for all x in [a,b]

I have a function, with domain (a,b) --> R, which is monotone (which impies it is bounded at each constant in (a,b)). I have two conditions that I am trying to meet

First, I am looking for an example of a monotone function with (a,b)-->R that is unbounded and then I need to verify that the function has lim_x-->c^+ less than or equal to Lim_x-->d^- whenever a < c < d < b