Functions : Convergence and Limits

Please see the attached file for the fully formatted problems. Let f be a real function defined by . 1) Evaluate f’(x), f’’(x), f(0). Show that f has exactly two roots and , with . Find an interval of two consecutive real numbers within which the roots must lie. From now on, let us denote and these two (closed) intervals. 2) Let be the sequence defined by and a) Show that for all whole natural numbers . b) Show that if the sequence is convergent, its limit is . c) Evaluate g’(x). Show that the sequence is convergent and give a sufficient number of iterations N such that the approximation error is no more than . 3) Let and let be the sequence defined by Newton’s... click for more

Sequence of Polynomials

Questions on a Sequence of Polynomials. See attached file for full problem description.

Solutions of Linear Equations

I would like a short explanation of Gaussian Elimination with partial pivoting and Gauss-Seidel. Also, explain when each applies or when one is better than the other. Please include some examples.

Equivalence Relation

Relation S is defined as followed: xSy iff y-x is an integer (x,y € R) where R=all real numbers a) prove that S is an equivalence relation on R b) Which real numbers belong to [-17]?

Doolittle LU Decomposition

Please see attachment. C(A) stands for C'infinite'(A).