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Real Analysis

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

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: Proof Regarding Bounded Subsets

Let A subset or equal of R be bounded above and let c belong to R.Define the sets c+A and cA by c+A={c+a : a belong to A} and cA={ca : a belong to A}. 1-show that sup(c+A)=c+ Sup A 2-if c>=0,show that sup(cA)=cSupA 3-postulate a similar type of statement for sup(cA)for the case c<0.

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

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

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.

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

Convergence of Series

Determine whether the series Sum(n!/n^n) n=1..infinity is absolutely convergent, conditionally convergent or divergent.

Series Test

Test the series (in the attached file) for convergence or divergence by using the Comparison Test or the Limit Comparison Test.

Real Analysis : Finding a Maximum using Lagrange Multipliers

Please see the attached file for the fully formatted problem. What is the maximum of F = x1 +x2 +x3 +x4 on the intersection of x21 +x22 +x23 + x24 = 1 and x31+ x32+ x33+ x34= 0? As this is an analysis question, please be sure to be rigorous and as detailed as possible.

Real Analysis : Mean Value Theorem

Let f(x) be integrable on [a,b], and let g(x) be nondecreasing and continuously differentiable on [a,b]. Let {p be element of P} be a partition of [a,b], and define U(f,g,p) = SIGMA (Mi(g(the ith term of x) - g(the (i-1)th term of x))) as i=1 to n L(f,g,p) = SIGMA (Ni(g(the ith term of x)-g(the (i-1)th term of x))) as i=1 t

Radius of Convergence

The problem is to determine the radius of convergence of the Taylor Series for each of the functions below centered at x. We are to explain our conclusion in each case. I would like to see how to work each problem (including what the Taylor Series is) and what the explanation is. a) centered at and NOTE: I know the