### Infinity Limit Functions

Find the limit(as R goes to infinity) of (-1/2+(1/4+V/f*R)^1/2)/(V/f*R)

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Find the limit(as R goes to infinity) of (-1/2+(1/4+V/f*R)^1/2)/(V/f*R)

Prove the sequence defined by a1 = 0, a2=1 and a(n+2)= (a(n+1)+a(n))/2 for n>=0, converges and find the limit. (See attachment for full question)

Prove that the open interval...with a, b being real numbers is an open set. (See attachment for full question)

Prove if... and... are bounded sequences of real numbers, then lim sup... (See attachment for full question)

Find the limit lim 3x/x+2 x->-2+

Lim x->4 x^2-16/x^4-256

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Complex Variables Power Series (I) Abel's Theorem: ∞ If the power series ∑ an zn converg

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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?

(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

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)

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.

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)

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

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.

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].

29.2 Prove that |cos x - cos y| ≤ |x - y| for all x, y in R

Show that arctan x<x for all x>0. (Hint: Look at the function f(x)= x - arctanx.)

Determine the radius of convergence and the exact interval of convergence of the following power series... Please see attached.

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.

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.

Show that there exists xE(0,π) with sinx = sin10x.

Please see the attached file for the fully formatted problems.

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).

Determine the limits of the following sequences if they exist. Justify your answers. (See attachment for full question)

Find a convergent subsequence and its limit of the attached sequence. State a tangible reason why your subsequence converges to the claimed limit.

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.

If (a_n)->0 and Absolute value of b_n -b<=a_n then show that (b_n)->b.