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Real Analysis

Convergence or Divergence, Taylor Polynomials, Maclaurin Series and Chain Rule

1. Test for convergence or divergence, absolute or conditional. If the series converges and it is possible to find the sum, then do so {see attachment} 2. Find the open interval of convergence and test the endpoints for absolute and conditional convergence: {see attachment} 3. For the equation f (x) = ... {see attachment

Evaluating a Limit

Thank you in advance for your help; this one sounds like it should be simple, but I still continually get the wrong answer: "Evaluate the limit as x goes to infinity of (1+(3/x))^(4x)."

Real Analysis : Proofs - Uniform Continuity

S(1): Let &#949;=1, and let any &#948;>0 be given. S(2): Let n be an integer > max(1, 1/&#948;), and set x=1/n and y=1/(n+1). S(3): Both x and y belong to (0,1), and |x-y| = 1/n(n+1) < 1/n < &#948;. S(4): However, |f(x)-f(y)| = |n-(n+1)| = 1 = &#949; S(5): This contradicts the definition of uniform continuity (i.e.,

Real Analysis : Fold Lines

By an n-fold line subdivision of the plane P, we mean any collection of n-distinct (infinite) lines in P, together with the open regions in P that they determine. (We don't count the lines as part of the regions.) Let us say that two such regions are adjacent if their boundaries have a positive-length or infinite line segment

Real analysis

Give formal negations of the following definitions: * Limit point. Your answer should be in the form: "A point p in X is NOT a limit point of the set E in X if ... " * Interior point. Your answer should be in the form: "A point p in X is NOT an interior point of the set E in X if ... " * Closed set. Your answer

Real Analysis - Open Intervals

Fix a point p in R. Let { I&#945; } be a ( possibly infinite ) collection of open intervals I&#945; = ( c&#945; , d&#945; ) which is a subset of R, such that p&#1028; I&#945; for all &#945;. Prove that the union I: = U&#945; I&#945; is also an open interval ( possibly infinite ). Hint: Cons


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

Real Analysis...Open Sets

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


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


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