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# Real Analysis

### 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

### Infinite Sequence : Convergence, Divergence and Sums (4 Problems)

1. Determine whether the sequence {see attachment} converges, and find its limit if it does converge ... 2. Determine whether the given infinite series converges or diverges. If it converges, find its sum ...

### 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)&#8319; b

### Limits - 3 Problems

Find each limit or explain why it does not exist. Please see attachment.

### 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

### It is an explanation of the Cauchy-Hadamard theorem on power series. Cauchy- Hadamard Theorem :- &#8734; For every power series &#8721; anzn there exist a number R, 0 &#8804; R < &#8734; n=0 called the radius of convergence with the following properties: (i) The series converges absolutely for every |z| < R (ii) If 0 &#8804; &#961; < R, the series converges uniformly for |z| &#8804; &#961; (iii) If |z| > R, the terms of the series are unbounded and the series is consequently divergent.

Complex Variable Cauchy- Hadamard Theorem :-

### 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

### Show that, is open then is open.

Real Analysis Let and let be continuous map given by .

### 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

### Let V be a region in &#8299;3complying with the hypotheses of the divergence theorem, and denote by S its boundary surface. Let also &#966;: &#8594; &#8299; be a scalar function, and c an arbitrary constant vector. By applying the divergence theorem to the vector field &#966;c (1) show that: (&#8747;&#8747;&#8747;v &#9660;&#966;dV - &#8747;&#8747;s &#966;ndS).c = 0 with the understanding that the integral of a vector is the vector of the integrals of the components. (2) Use the above result to deduce carefully that: &#8747;&#8747;&#8747;v &#9660;&#966;dV = &#8747;&#8747;s &#966;ndS.

Real Analysis Divergence Theorem Let V be a region in &#8299;3complying with the hypotheses of the divergence theorem, and denote by S its boundary surface. Let also &#966;: &#8

### 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)

### Analysis - Limit of the Average of the first N terms of a Sequence

Proving in Real Analysis Course. Please see the attached.

### 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 : Limits of Bounded Sequences

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

### 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?

### 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 : 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 : 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

### Real Analysis : Twice Differentiable Functions

Let g:[0,1]->R be twice-differentiable (i.e both g and g' are differentiable functions) with g''(x)>0 for all x belong to [0,1].if g(0)>0 and g(1)=1 show that g(d)=d for some point d belong to (0,1) if and only if g'(1)>1.

### Real Analysis: Points on a Differentiable Function

Let h be a differentiable function defined on the interval [0,3], and assume that h(0)=1 h(1)=2 and h(3)=2. a- argue that there exists a point d belong to [0,3] where h(d)=d. b-argue that at some point c we have h'(c)=1/3. c-argue that h'(x)=1/4 at some point in the domain.

### Real Analysis : Contractiveness

Prove that a function f is contractive on a set A if there exists a constant 0<s<1 such that Absolute value of f(x)-f(y)<=s*Absolute value of x-y for all x,y belong to A.show that if f is differentiable and f' is continous and satisfies Absolute value of f'(x)<1 on a closed interval then f is contractive on this set.

### Real analysis : Lipschitz Criterion

A function f:A->R is Lipschitz on A if there exists an M>0 such that Absolute value of f(x)-f(y)/x-y <=M for all x,y belong to A. show that if f is differentiable on a closed interval [a,b] and if f' is continous on [a,b] then f is Lipschtiz on [a,b]. Geomtrically speaking, a function f is Lipschitz if there is a uniform bound

### Real Analysis : Cauchy Criterion for Uniform Convergence

Prove that A sequence of functions (f_n) defined on a set A subset or equal to R converges uniformly on A if and only if for every e>0(epsilon) there exists an N belong to N such that Absolute value of f_n (x)-f_m (x)<e for all m,n>=N and all x belong to A.