Integration and Limits Proof

If a>0 show that pi lim ∫ sin(nx)/nx dx = 0 a What happens if a = 0. 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. ...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 : Convergence and Changes to a Finite Nuber of Terms

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.

Differentiable Functions : Lipschitz and Absolute Values

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]

Limits and Monotone Functions

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 keywords: monotonic

Prove that an increasing real-valued function f which is defined on an open interval has at most countably many points c at which f(x) does not converge to f(c).

Prove that if f is an increasing real-valued function on an open interval (a, b), then, for all but at most countably many points c in (a, b), Lim_(x-->c) f(x) exists and is equal to f(c).

Continuous Functions, Sequence of Functions and Convergence

Please see the attached file for the fully formatted problem.

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.

Real Analysis : 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(x)fj(x)dx has determinant zero. a See attached file for full problem description.