Real Analysis Answers

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)

find the fourier sine series for the function f(x)=e^ax for 0<x<pi where a is a constant

Given a sequence {a_n} with a_n part of X and X is a metric space. Prove that if a_n is convergent, then the limit is unique.

Consider a number a in the Reals with a>0 and n in the Naturals. Show that there exists a unique x in the Reals such that x^n=a

1. Consider the function ( ) 2 3 f x  x 
a. Show that f (x) has a root in the interval [1, 2].
b. Compute an approximation to the root by taking 4 steps of the bisection method.(BY HAND)
c. Repeat, 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)

n(-N: (1/1)*2+(1/2)*3+...+(1/n)(n+1)=n/(n+1)

Use the definition to prove that the set of all odd natural numbers is a countable set

Use Bisection method to calculate the first root for :
f(d)= 257d^2-640d=0
a=0.0000 b=5.0000
Tolerance=0.0500

Let Xn be a bounded sequence of real numbers. Prove that the minimal subsequential limit of Xn is computed by the formula
supK (inf(n>=K) An)

Hint: Prove first that it is a subsequential limt.