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Real Analysis

Real Analysis lim sup and lim inf

Let {a_n} and {b_n} be sequences in [-infinity,+infinity] and prove the following assertions: 1). a).Lim sup (as n -> infinity) ( a_n + b_n) less than or equal to lim sup a_n + lim sup b_n ( as n foes to infinity). b).Show by an example that strict inequality can hold. Provided none of the sums is of the form infin

Radius of convergence in series (complex plane)

1). Find the radius of convergence for each of the following power series. Please check my solution for this problem: a). sum ( n = 0 to infinity) a^n z^n, a is a complex number. My solution: R( radius of convergence) = lim |a_n/a_n+1) = lim | a^n/a^(n+1)| = 1/|a| b). Sum ( n=0 to infinity) = lim|a^(n^2)*z^n, a is

Real Analysis Topology and Sigma-Algebra

1). Prove that any sigma-algebra, which contains a finite number of members is also a topology. ( The Q in another words : to show that there exist a sequence of disjoint members of a sigma algebra which contains infinite no. of members). 2). Does there exist an infinite sigma-algebra which has only countably many members?

Real Analysis : Sigma-algebra

Suppose X is a measurable space, E belongs to the sigma algebra ( I believe to the sigma algebra in X) , let us consider XE = Y. Show that all sets B which can be expressed as AE, where A belongs to the sigma algebra in X, form a sigma-algebra in Y. Please justify every step and claim you make in the solution.

Effects of Inflation and Deflation and Types of Annuities

1) How would economic pressures like inflation or deflation affect your decision to make a long term investment? Should our mathematical analysis take these factors into consideration? 2) What are some different types of annuities that you have used or heard of?

Finding the Sum of a Harmonic Series

The series .... 1 + 1/2 + 1/3 + 1/4 + 1/6 + 1/8 + 1/9 + 1/12 + .... Where the terms are the reciprocals of the positive integers whose only prime factors are 2's and 3's. What is the sum of the series?

Real Analysis : Limits and Cluster Points

Please see the attached file for the fully formatted problems. 1) Prove that does not exist but that . 2) Let f, g be defined on to , and let c be a cluster point of A. Suppose that f is bounded on a neighborhood of c and that . Prove that . 3) Let f, g be defined on A to and let c be a cluster point o

Taylor Polynomial

Use Taylor polynomials about 0 to evaluate sin(0.3) to 4dp,showing all workings. 1)F(x)=square root 4+x and G(x)=square root 1+x by writing square root of 4+x=2 square root 1+1/4x and using substitution in one of the standard Taylor series, find the Taylor series about 0 for f.Given explicitly all terms up to term in x raise

Sequences and limit boundaries

1) If (bn) is a bounded sequence and lim(an) = 0, show that the lim(anbn) = 0. Explain why Theorem 3.2.3 cannot be used. Note: Here's Theorem 3.2.3 (a) Let X = (xn) and Y = (yn) be sequences of real numbers that converge to x and y respectively, and let c be an element R. Then the sequences X+Y, X-Y, X∙Y, and cX co

Project Evaluation Review Technique and Critical Path Method

Please refer to the attached file for this PERT / CPM problem: I've determined that the critical path for this network is A - E - F and the project completion time is 22 weeks. Here's where I need help: If a deadline of 17 weeks is imposed, what activities should be crashed?

Pin-Joint Analysis : Direction and Magnitudes of Support Reactions

I need a worked solution for these questions (also attached on the LAST PAGE of the attachement. The first couple pages are just examples. Thanks) The structure shown in Figure TA 1 is a pin-jointed section of a canopy and carries a single load of 4 kN acting at the lower right-hand joint. [DIAGRAM] Pin-jointed canopy De

Infinite Series : Convergence and Divergence (8 Problems)

1) a) Prove that N ∑ 1/n(n+1) = 1- (1/N+1) n=1 Hence, or otherwise, determine whether the following infinite series is convergent or divergent: b) Determine whether each of these infinite series are convergent or divergent. Justify your an

Taylor Expansion, Interval of Convergence and Calculation of 'pi'

In 1671, James Gregory, a Scottish mathematician, developed the following series for tan^-1 x {See attachment} 1. Verify that Gregory's series is correct by using a Taylor Series expansion or methods of power series. 2. Find the interval of convergence of Gregory's series. 3. Using Gregory's series, find a series whose

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

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

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

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