### 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.

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

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?

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

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))

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

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

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.

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

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

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.

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

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

Prove that the sequence defined by X1=3 and X_n+1=1/(4-X_n) converges

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

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

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

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

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

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

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

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

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

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

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

Limit x to 9 Square root of x - 3(3 is not part of the square root)/x-9

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

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.

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

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

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?