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Real Analysis

Infinite Series, Convergence and divergence test

PLEASE SHOW ALL WORK, STEP-BY-STEP, WITH ALL CORRECT NOTATION. My notation is lousy for some reason the attachment won't come through, so I cut and pasted it below. Determine whether the following Diverge (D), Converge Conditionally (CC), or Converge Absolutely (AC). Give the rationale for each response. Must show all work

Real analysis

Prove: subsequences of a convergent sequences converge to the same limit as the original sequence

Real Analysis

Show that lim inf a_n = lim sup a_n if and only if lim a_n exists

Real Analysis

Show that if x_n <= y_n <= z_n for all n belong to N and if lim x_n=limz_n=L then lim y_n=L as well

Real Analysis

Let (a_n) be a bounded sequence and assume lim b_n=0.show that lim(a_n b_n)=0.

Real Analysis

A) Show that if (b_n)-->b,then the sequence of absolute values Absolute value of b_n converges to absolute value of b

Real Analysis

Show that limits, if exist, must be unique. In other words, assume lim an=L1 and lim an=L2 and prove that L1=L2

Real Analysis

Let xn(smaller n)>=0 for all n belong to N a) if (xn)-->0, show that(sqrt[xn])-->0 b)if (xn)-->x,show that(sqrt[xn])-->x

Real Analysis

Using the definition of convergence of a sequence show that the following sequences converge to the proposed limit: 1-lim 1/(6n^2+1)=0 2-lim 2/sqrt[n+3]=0 3-lim (3n+1)/(2n+5)=3/2

Real Analysis

1- if A1,A2,A3,...,Am are each countable sets, then the union A1 U A2 U A3...U Am is countable 2- if An is a countable set for each n belong to N,then Un=1(to infinity) An is countable

Real Analysis

Proof: if A subset or equal of B and B is countable, then A is either countable, finite or empty.

Real analysis

Prove that if a is an upper bound for A and if a is also an element of A, then it must be that a=sup A

Real Analysis

Assume that A And B are nonempty, bounded above and satisfy B subset or equal of A. Show that sup B<= sup A

Real Analysis

If sup A < sup B then show that there exists an element b belong to B that is upper bound for A.

Real Analysis

Let y1=1,and for each n belong to N define y_n+1=(3y_n+4)/4. a-Use induction to prove that the sequence satisfies y_n<4for all n belong to N. b-use another induction argument to show the sequence(y1,y2,y3,...)is increasing.

Real Analysis

Given a function f and a subset A of its domain, let f(A) represent the range of f over the set A; f(A)={f(x) : x belong to A}. a-Let f(x)=x^2. If A=[0,2](the closed interval{x belong to R : 0<=x<=2}) and B=[1,4],find f(A) and f(B).Does f(A intersection B)=f(A) intersection f(B) in this case?.Does f(A U B)=f(A) U f(B)?. b-Fi

Real analysis : Measurable Functions

Q1. If 0<=a_1<=a_2<=a_3<=....,( 1,2,3 are the subscripts of a) 0<=b_1<=b_2<=b_3<=......(1,2,3 are the subscripts of b) and a_n --> a and b_n -->b Then prove that a_n*b_n -->a*b Q2.Let f: R --> R be monotonically increasing, i.e. f(x_1)<= f(x_2) for x_1< = x_2. Show that f is measurable. Hint: You may extend f to f':[-in

Radius of Convergence : Summations and Power Series

Please see the attached file for full problem description. a) Evaluate b) What is the radus of convergence .... .... To what simple function does this series converge? c) Is f(z)=... ... analytic near z= -1? and expand f(z)= ... in a power series near z= -1 can we predict the domain of convergence from the outset?

Real Analysis

Please see the attached file for the fully formatted problems. Suppose that is not a perfect nth power, i.e K is not equal to (a) Prove that is not a member of Q, the set of all rational numbers. (b) Infer that the nth root of a natural number is either a natural number or it is irrational.

Convergence

Please see the attached file for full problem description.

Limits

I attached a word document. Be sure to show me all of your work so that I can fully understand how to do the problems correctly. Thank you very much for your help.

Real-Life Application : Examples of Data Modeled Using a Linear Formula

Find an article through newspapers, magazines, professional journals, etc and find at least two examples of data that are best modeled using linear formulae. Describe the importance of each example and why a linear model is appropriate for the data. Note that we are referring to a linear model not simply a time chart where dots

Riemann Sum and Limit

Write out the Riemann Sum for R(f,P, 0, 2) for arbitrary n, f(x) = x2&#8722;3x+2, where each &#8710;xk = 2/n and ck = xk, simplify and use the formulas &#8721;n,k=1 k=(n(n+1))/2 and &#8721;n,k=1 k2=n((n + 1)(2n + 1))/6 to find the limit as n --> 1.

Limits

See attached lim (sin^2 3t)/2t t--->0