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Real Analysis

Limit

Lim sqrt (x+3) - sqrt (3)/x x->o

Rootfinding for Nonlinear Equations

24. Which of the following iterations will converge to the indicated fixed point alpha (provided x_0 is sufficiently close to alpha)? If it does converge, five the order of convergence, for linear convergence, give the rate of linear convergence. a) x_n+1 = -16 + 6x_m + 12/x_n alpha = 2 b) x_n+1 = 2/3x_n + 1/(x_n)^2

Using a Summation Series to Estimate a Quantity

Say the only tool given to you is a calculator which performs addition, subtraction, multiplication, and division. Let X= Summation (k=1 -->n) e^-(k/n)^2 with N^20 Explain a practical way of computing X within an error of 10^8. Roughly how big is X?

Covers and Convergent Series

(4) (a) Let I1,I2,I3... be open intervals and let J be a closed interval and let J be a closed inteval. Let lk be the length of Ik, and let L be the length of J....Please see the attachement

Real Analysis : Subintervals

Prove rigorously: Let N be an integer > or equal to 2, and let Xsub0....Xsubn E [0,1). Prove that there exist i and j with i not equal to j such that abs (xsubi-xsubj) < 1/n.

Real Analysis : Bounded Continuity / Differentiability

Problem: Let f: [0, &#8734;) &#8594; R be a bounded function. For all X greater than or equal to 0, let G(x)=sup{f(t): 0 is less than or equal to t is less than or equal to x} a) Show that if f is continuous, g is also continuous. Is the converse also true? Justify. b) If f is differentiable and continuous, is g also d

Real Analysis

29.18 Let f be a differentiable on R with a = sup {|f &#8242;(x)|: x in R} < 1. Select s0 in R and define sn = f (sn-1) for n &#8805; 1. Thus s1 = f (s0), s2 = f(s1), etc Prove that (sn) is a convergence sequence. Hint: To show (sn) is Cauchy, first show that |sn+1 - sn| &#8804; a&#1468;|sn - sn-1| for n &#8805; 1.

Real Analysis

29.12 (a) Show that x < tan x for all x in (0, &#960;/2). (b) Show that x/ sin x is strictly increasing function on (0, &#960;/2). (c) Show that x &#8804; (&#960;/2)&#1468;sin x for all x in [0, &#960;/2].

Real Analysis

29.2 Prove that |cos x - cos y| &#8804; |x - y| for all x, y in R

Real Analysis : Constant Function

Let f be defined on R and suppose it satisfies |f(x + y) - f(x)| &#8804; |y|^(3/2) for all x, yER. Show that f is a constant function.

Real Analysis : Bounded Sets

Please see the attached file for the fully formatted problem. Let S be a bounded nonempty set and let S^2 = {s^2 : s E S}. Show that sup S^2 = max((sup S)^2, (inf S)^2).

Real Analysis : Limit Superior

Let a_n be bounded sequence.prove that a-the sequence defined by y_n=sup{a_k:k>=n} converges. b- Prove that lim inf a_n<=lim sup a_n for every bounded sequence and give example of a sequence which the inequality is strict.

Real Analysis : Neighborhoods

Assume g:(a,b)->R is differentiable at some point c belong to (a,b). If g'(c)not= 0 show that there exists a delta neighborhood V_delta (c) subset or equal to (a,b) for which g(x) not= g(c) for all x belong to V_delta (c).

Real Analysis : Differentiable and Increasing Functions

A-a function f:(a,b)->R is increasing on (a,b) if f(x)<=f(y) whenever x<y in (a,b). Assume f is differentiable on (a,b). Show that f is increasing on (a,b)if and only if f'(x)>=0 for all x belong to (a,b). b-show that the function g(x){x/(2+x^2 sin(1/x)) if x not=0 0 if x=0 is differentiable on R and satisfies g'(0)>0.Now

Real Analysis : Twice Differentiable Functions

Let g:[0,1]->R be twice-differentiable (i.e both g and g' are differentiable functions) with g''(x)>0 for all x belong to [0,1].if g(0)>0 and g(1)=1 show that g(d)=d for some point d belong to (0,1) if and only if g'(1)>1.

Real Analysis : Differentiability

Prove that if f and g are differentiable functions on an interval A and satisfy f'(x)=g'(x) for all x belong to A, then f(x)=g(x)+k for some constant k belong to R.

Real Analysis : Uniform Convergence

Let (f_n) be a sequence of diffrentiable functions defined on the closed interval [a,b] and assume (f'_n) converges uniformly on [a,b]. Prove that if there exists a point xo belong to [a,b] where f_n(xo) is convergent, then (f_n) converges uniformly on [a,b].

Real analysis

G(x)=Sum sign(m top n=0 bottom)(1/2^n)h(2^n x).for more inf. please check #30026,#30028,#30029. show that (g(x_m)-g(0))/(x_m - 0)=m+1, and use this to prove that g'(0) does not exist. any temptation to say something like g'(0)=oo should be resisted. setting x_m=-(1/2^m) in the previous argument produces difference heading to

Real analysis

Taking the continuity of h(x) as given in#30026,#30028 by using any of the functional limits and continuity theorems prove that the finite sum g_m (x)=sum sign(oo top n=0 bottom) of 1/2^n h(2^n x) is continous on R