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Real Analysis

Real Analysis : Continuity, Closed and Open Sets and Differentiability

Prove OR disprove the following statements. Explain. (i) There is a nonempty set S in R ( real numbers) such that S contains none of its limit-point(s). (ii) There is a nonempty set S in R such that S is neither open nor closed. (iii) There is a nonempty set S in R such that S is both open and closed. (iv) Let a

Real Analysis : Differentiability and Sequence of Partial Sums

A). Prove that the function f(x) = e^x is differentiable on R, and that (e^x)' = e^x. ( Hint: Use the definition of e^x, and consider the sequence of partial sums.) My thoughts on a: I tried to prove the differentiability by proving continuity on R, since e^x is series, sum of polynomials, and all polynomials are different

Closure, Convergence, Differentiability, Integrability, Sequences

Prove or disprove the following statement. Provide detailed answer and justify all steps. 1). There is a nonempty set S in R such that closure of S is equal to R and the closure of its complement closure(R-S) also is equal to R. My thoughts on this problem: Q ( rationals) and R-Q (irrationals), but how to prove that the cl

Sequences : Limits and Convergence

This question is from Advanced Calculus II class, it is more like introduction to real analysis. Let f_n: R -> R be the sequence of functions given by f_n(x) = x/ ( 1 + nx^2) a). Prove that the sequence f_n converges uniformly to a function f. What is f? b). Prove that for each x in R-{0} ( all real numbers but 0 not in

Real Analysis : Lebesgue Integral Problem

Let f be a nonnegative measurable function. Show that (integral f = 0) implies f = 0 a.e. See attached document for notations. Please help: This problem is from Royden's Chap 4 text on Lebesgue Integral.

Real Analysis : Lebesgue Measure

Show that the sum and product of two simple functions are simple. Show that [Definition of simple: A real-valued function is called simple if it is measurable and assumes only a finite number of values. If is simple and ahs the values then , where .] This problem is from Royden's Real Analysis text for gradu

Central limit theorem for a company

Let's say you work in a company where over 2000 people are employed. Using the Central Limit Theorem, where the mean age of all employed is 37 with a standard deviation of 13; If 5 people are randomly selected, find the probability of their age being less than 22.

40 Problems : Sequences, Series, Convergence, Divergence and Limits

1. For each of the sequences whose nth term is given by the formula below (so of course n takes successively the positive integer values 1,2,3...), does it have a limit as n tends to infinity? In each case, briefly explain your answer including justification for the value of the limit (if it exists) a) (1/3)ⁿ b

Taylor & Maclaurin Series

1- Find the Taylor series generated by f at X = a. f (x) = 1/(10-x) a = 3 2- Find the Maclaurin series for the given functions. A) 1/(6+x) B) sin 10x.

Power Series; Sum of Series; Estimate Using Terms

Please assist me with the attached problems, including: 8.7 Find the convergence set for the power series ... 8.8 Given the series (a) estimate the sum of the series by taking the sume of the first four terms. How accurate is the estimate? (b) How many terms of the series are necessary to estimate its sume with three-place

Vectors, Maclaurin and Taylor Series and Radius of Convergence

Please assist me with the attached problems, including: 1. Sketch each vector 2. Find the standard representation and length of each vector 3. Maclaurin and Taylor Series and Radius of Convergence See attachment for complete list of questions. Thanks.

Average Value of Continuous Functions and Limits

The definition of average value of a continuous function can be extended to an infinite interval by defining the average value of f on the interval [a, ∞) to be Lim as t approaches ∞ 1/(t-a)integrand from a to t f(x)dx 1. Find the average value of {see attachment} on [0, ∞). 2. Find the lim as x goes t

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

Poles, Taylor Series, Laurent Series and Power Series

Suppose that a function F(s) has a pole of order m at s=s0, with a Laurent series expansion ... in the punctured disk 0<|s - s0|<R2, and note that (s-s0)&#8319; F(s) is represented in that domain by the power series ... By collecting the terms that make up the coefficient of (s-s0)&#8319;­&#1471;¹ in the product (Sec. 61) of

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)."

Open Cover

Real Analysis Let xEn/n be an irrational number. For each rational number yEn define... Let be the collection of all these open balls of n. Is α an

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

Continuity and limits points

1. For i = 1,2 let fi: Xi --> Yi be maps between topological spaces. Show that the product f1Xf2: X1XX2 --> Y1XY2 defined by f1Xf2(x1x2):= (f1(x1), f2(x2)) is continuous if and only if f1 and f2 are continuous. *(Please see attachment for proper representation of formulas and problem #2)

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

Prove that union is an open interval.

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