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Real Analysis

A Limit

Find the following: the limit as x approaches 0 of (tan 3x * cot 2x)

Real Analysis

A set F subset or equal to R is closed if and only if every Cauchy sequence contained in F has a limit that is also an element of F.

Real Analysis

Show that if sum x_n converges absolutely and the sequence(y_n) is bounded then the sum x_n y_n converges.

Real Analysis

1-Show that if sum a_n converges absolutely then sum a^2_n also converges absolutely.Does this proposition hold without absolute converge. 2-if sum a_n converges and a_n>=0 can we conclude anything about sum of sqrt a_n?

Real Analysis

Let Sum of an(sign of sum) be given.For each n belong to N let p_n=an if a_n is positive and assign p_n=0 if a_n is negative.In a similar manner,let q_n=a_n if an is negative and q_n=0 if a_n is positive. 1-Argue that if Sum a_n diverges then at least one of sum p_n or sum q_n diverges. 2- show that if sum a_n converges co

Power series multiplication

Please show proper notation, justification and step by step work. n See attachment for problem Given that the zeros for (sinx)/x are the values x=0, x= +-pie, x=+-2pie, x=+-3pie,.... (x=0 must be excluded, why?) This implies that F(x) can be factored as follows F(x) = (1-(x/pi)) (1-(x/-pi)) (1-(x/2pi)) (1-(x/3pi))

Real Analysis

Show that if the series sum(sum sign) to infinity(top) of k=1(bottom) of a_k converges then a_k--->0

Real Analysis

Show that if sum(sum sign) to infinity(top) of k=1(bottom) of a_k=A and sum(sum sign) to infinity(top) of k=1(bottom) of b_k=B, then 1-sum(sum sign) to infinity(top) of k=1(bottom) of ca_k=cA for all c belong to R 2-sum(sum sign) to infinity(top) of k=1(bottom) of (a_k+b_k)=A+B

Real Analysis

Assume a_n and b_n are Cauchy sequences.Use a triangle inequality argument to prove c_n=Absolute value of a_n-b_n is Cauchy.

Real Analysis

Show that the series sun sign over it infinity sign and below it n=1 of 1/n^p converges if and only ifp>1

Real Analysis

Let (a_n) be a bounded sequence and define the set S={x belong to R: x< a_n for infinitely many terms a_n}. show that there exists a subsequence(a_nk) converging to s=sup S

Real Analysis : Converging Sequences

Assume (a_n) is a bounded sequence with the property that every convergent subsequence of (a_n) converges to the same limit a belong to R.show that (a_n) must converges to a.

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