Real Analysis Answers

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.

Suppose that K1,K2 c R^n are nonempty compact sets. Let
d(K1;K2) = inf{d(x1, x2)| (x1 cK1) and (x2c K2)}:
Show that d(K1,K2) = 0 implies( K1 n K2) is not empty set.

1. Consider the function f(x)=x^3-2 using Newton's method. Take x0=1.5 for the starting value.
using Newton's method. Take x0=1.5 for the starting value.
For each method, present the results in the form of table:
Column1:n (step)
Column2:xn (approximation)
Column3: ( ) n f x
Column4:| xn - xn-1 | (error)