Infinitely Differentiable Function that's not Analytic

Use the given information: the functions g:[a,b]->R and h:[a,b]->R are continuous with h(x) >= 0 for all x in [a,b], and there is a point c in (a,b) such that: the integral from a to b of h(x)g(x)dx = g(c) times the integral from a to b of h(x)dx. to show that the Cauchy Integral Remainder Theorem implies the Lagrange Remainder Theorem if the function f^(n+1):I->R is assumed to be continuous.

Cauchy Integral Remiander Theorem Applied to Newton's Binomial Expansion

Apply the Cauchy Integral Remiander Theorem in the analysis of the expansion (Newton's binomial expansion): ln(1+x) = the sum from k=1 to infinite of (-1)^(k+1) times (x^k/k) if -1

Weierstrass Approximation Theorem

This question is about the Weierstrass Approximation Theorem Show that the Approximation Theorem does not hold if we replace I by R(real number system), by showing that if f(x) = e^x for all x, then f:R->R cannot be uniformly approximated by polynomials.

Weierstrass Approximation Theroem

Note: abs = absolute value Define f(x) = abs(x - 1/2) for 0 <=x <= 1. Use the proof of the Approximation Theorem to find an explicit polynomial p:R->R such that abs(f(x) - p(x)) < 1/4 for all x in [0,1]

Solving for n as a Factorial

Does there exist a natural number n such that [n!/(n-4)!] = 11,880 ? if so find n, if not explain why not (Hint: factor 11,880 into its prime factors)