### Real Analysis : Convergent and Divergent Summations

Give an example to show that its possible for both Sum of x_n(sum sign) and sum of y_n to diverge but for Sum of x_n y_n to converge.

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Give an example to show that its possible for both Sum of x_n(sum sign) and sum of y_n to diverge but for Sum of x_n y_n to converge.

Let f and g be functions defined on a domain A subset or equal to R, and assume lim_x-->c f(x)=L and lim_x-->c g(x)=M for some limit point c of A then, 1-lim_x-->c k f(x)=kL for all k belong to R. 2-lim_x--> [f(x)+g(x)]=L+M 3-lim_x-->c [f(x)g(x)]=LM 4-lim_x-->c f(x)/g(x)=L/M provided M not = 0

If {G1,G2,G3,...} is a countable collection of dense, open sets then the intersection (U top infinity bottom n=1)G_n is not empty.

Definition: A set A subset or equal to R is called an F_&(F sigma) set if it can be written as the countable union of closed sets. A set B subset or equal to R is called G_&(G sigma) if it can be written asthe countable intersection of open sets. 1-Argue that a set A is a G_& (G sigma) set if and only if its complement is

A set E is totally disconnected if, given any two points x,y belong to E there exist separated sets A and B with x belong to A and y belong to B and E=A U B. 1-show that Q is totally disconnected. 2-is the set of irrational numbers totally disconnected?

Let A and B be subsets of R show that if there exists disjoint open sets U and V with A subset or equal of U and B subset or equal of V then A and B are separated.

Show that if K is compact and F is closed then K intersection F is compact.

Show that if K is compact, then sup K and inf K both exist and are elements of K

Let A be bounded above so that s= sup A exists show that s belong to closure A(A over it bar)

Let x belong to O, where O is an open set.If (x_n) is a sequence converging to x prove that all but a finite number of the terms of (x_n) must be contained in O.

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

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.

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