Real Analysis : Limits of Bounded Sequences
Prove if... and... are bounded sequences of real numbers, then lim sup... (See attachment for full question)
Real Analysis
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Limits Found
Find the limit lim 3x/x+2 x->-2+
limit function
Lim x->4 x^2-16/x^4-256
Solving the Limit Function
Lim sqrt(x+7)-9/ x-74 x-> 74
Limit Explanation Solved
Lim sqrt (x+3) - sqrt (3)/x x->o
Power Series (I): Abel's Theorem
Complex Variables Power Series (I) Abel's Theorem: ∞ If the power series ∑ an zn converg
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 -- Real Numbers/Integers Theory
Let x be a real number, and let N be an integer ≥ 2. Prove that there exist integers P and Q such that: 1 ≤ q ≤ N and absolute value of [x-(P/Q)] < 1/(QN)
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 Single-Variable Analysis Problem
Let x be an irrational number. Prove that there exist infinitely many fractions (p/q) with p and q as integers such that: abs(x-[p/q]) < 1/(q^2)
Real Analysis : Bounded Continuity / Differentiability
Problem: Let f: [0, ∞) → 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 ′(x)|: x in R} < 1. Select s0 in R and define sn = f (sn-1) for n ≥ 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| ≤ aּ|sn - sn-1| for n ≥ 1.
Real Analysis Trigonometric Function
29.12 (a) Show that x < tan x for all x in (0, π/2). (b) Show that x/ sin x is strictly increasing function on (0, π/2). (c) Show that x ≤ (π/2)ּsin x for all x in [0, π/2].
Real Analysis Proof Function
29.2 Prove that |cos x - cos y| ≤ |x - y| for all x, y in R
Real Analysis : Arctan X
Show that arctan x<x for all x>0. (Hint: Look at the function f(x)= x - arctanx.)
Real Analysis : Radius and Interval of Convergence- Power Series
Determine the radius of convergence and the exact interval of convergence of the following power series... Please see attached.
Real Analysis : Constant Function
Let f be defined on R and suppose it satisfies |f(x + y) - f(x)| ≤ |y|^(3/2) for all x, yER. Show that f is a constant function.
Real Analysis : Power Series - Expansion and Interval of Convergence
Let ... a) Show that f can be represented by a power series. What is its interval of convergence? b) Calculate the power series expansion for the function F(x) = ... Please see attached for equations.
Real Analysis : Theorem for Existence of a Root
Show that there exists xE(0,π) with sinx = sin10x.
Real Analysis : Convergence of Series (3 Problems)
Please see the attached file for the fully formatted problems.
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 : Limits Sequences
Determine the limits of the following sequences if they exist. Justify your answers. (See attachment for full question)
Convergent Subsequence and Limit of Sequence
Find a convergent subsequence and its limit of the attached sequence. State a tangible reason why your subsequence converges to the claimed limit.
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 : Squeeze Theorem
If (a_n)->0 and Absolute value of b_n -b<=a_n then show that (b_n)->b.
Real Analysis: Differentiability and Limits
Prove : Assume f and g are continous functions defined on interval contaning a, and assume that f and g are differentiable on tis interval with the possible exception of the point a. If f(a)=0 and g(a)=0 then lim f'(x)/g'(x)=L as x->a implies lim f(x)/g(x)=L as x->a.
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