Real Analysis Answers

Using the epsilon-delta definition verify that
lim x-->3 (x^2-x) = 6.

Using the epsilon-delta definition of the limit, prove that if lim x-->a f(x) and lim x-->a g(x) exist, then
lim x-->a [f(x) + g(x)] = lim x-->a f(x) + lim x-->a g(x).

Show that if lim a_n = negative infinity, the lim 1/a_n = 0.

prove that 7 divides 2^(3n)-1

Show that if lim a_n = positive infinity then lim (-a_n) = negative infinity.

Show that if lim a_n = -infinity, then lim 1/a_n = 0

A store advertises a circular Rug as being 4.9m2. Travis wants a rug to fit a rectangular space that is 2.6m by 2.6m. Will the rug fit?

Determine whether the series converges.
summation[1/{n+(-1)^n}^2]

a) Let f(x) be a C^1 function of period 2pi. Show that we may as well assume that f(0) = 0 and we need only show that the fourier series converges to zero at x = 0
b) Let g(x) = f(x)/(e^ix-1). Show that g(x) is a continuous function
c) Let Cn be the complex fourier coefficient of f(x) and Dn the coefficients of g(x). Show that Dn ---> 0
d) Show that Cn = Dn-1 - Dn so that the series sigma (Cn) is telescoping
e) deduce that the fourier series of f(x) at x = 0 converges to zero

Indefinite limit problem:
Show that If an < 0 and lim an = 0, then lim 1/an = negative infinity.