please i need the question (m) (n)(o) for the detailed proof and example ! thanks !
0 and limn→” an-L > 0, then (m) If an, bn lim an) (lim supbn) lim sup (anbn) - (n) If an > 0 and bn > 0 and if both lim supn→ooan and lim supn→oobn are either finite or infinite, then lim sup,-,(anbn) 0, then lim infn→oo atl oo an
Homework Answers
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Prove that the given statements are true concerning the two sequences of real numbers (an) and (b...
18. If ai, az, as,... is a bounded sequence of real numbers, define lim sup an...
18. If ai, az, as,... is a bounded sequence of real numbers, define lim sup an (also denoted lim an) to be --+ n+ l.u.b. {z ER: an > & for an infinite number of integers n} and define lim inf an (also denoted lim an) to be g.l.b. {ER: An <for an infinite number of integers n}. Prove that lim inf an Slim sup an, with the equality holding if and only if the sequence converges. 19. Let ai,...
can you please explain a and b thanks Fourier Analysis See are two finite sequences of...
can you please explain a and b thanks
Fourier Analysis
See are two finite sequences of complex numben 7. Suppose (an)- and (bn)1 Let Br= bn denote the partial sums of the series b with the conventicn 1 Bo=0. (a) Prove the summation by parts formula N-1 anbn aNBN- aM BM-1 (an+1-an)B n M n-M (b) Deduce from this formula Dirichlet's test for convergence of a series: if the partial sums of the seriesb are bounded, and fan} is a...
2.13.4 2.13.5 Show that lim supno (-X) = -(liminf ,-Xn). If two sequences {an) and {bn}...
2.13.4 2.13.5 Show that lim supno (-X) = -(liminf ,-Xn). If two sequences {an) and {bn} satisfy the inequality an <b, for all sufficiently large n, show that limsupan Slim sup bn and liminfa, <liminf bn. 100 2.13.6 Show that lim, 100 Xn = o if and only if lim sup.Xn = liminf xn = c. n-00 2.13.7 Show that if lim sup a n = L for a finite real number L and € > 0, then an >...
{x_n} and {y_n} are sequences of positive real numbers AC fn→oo > O, prove tha m...
{x_n} and {y_n} are sequences of positive real numbers
AC fn→oo > O, prove tha m in yn lim xn 0 implies lim yn_0
Let (xn) be a bounded sequence of real numbers, and put u = lim supn→∞ xn . Let E be the set consisting of the limits of all convergent subsequences of (xn). Show that u ∈ E and that u = sup(E). Form...
Let (xn) be a bounded sequence
of real numbers, and put u = lim supn→∞ xn . Let E be the set
consisting of the limits of all convergent subsequences of (xn).
Show that u ∈ E and that u = sup(E).
Formulate and prove a similar result for lim infn→∞ xn .
Thank you!
7. Let (Fm) be a bounded sequence of real numbers, and put u-lim supn→oorn . Let E be the set consisting of the limits of...
Let {an} m-o and {bn}ņ=be any two sequences of real numbers, we define the following: N...
Let {an} m-o and {bn}ņ=be any two sequences of real numbers, we define the following: N • For any real number L € R, we write an = L if and only if lim Lan = L. N-0 n=0 n=0 X • We write an = bn if and only if there is a real number L such that n=0 n=0 I and Σ. = L. Select all the correct sentences in the following list: X η (Α) Σ Σ...
Prove the following Definition 6.6.1 (Subsequences). Let (an) =) and (bn), m=0 be sequences of real...
Prove the following
Definition 6.6.1 (Subsequences). Let (an) =) and (bn), m=0 be sequences of real numbers. We say that (bn)is a subsequence of (an) a=iff there exists a function f :N + N which is strictly increasing (i.e., f(n + 1) > f(n) for all n EN) such that bn = f(n) for all n E N.
5. Let {xn} and {yn} be sequences of real numbers such that x1 = 2 and...
5. Let {xn} and {yn} be sequences of real numbers such that x1 =
2 and y1 = 8 and for n = 1,2,3,···
x2nyn + xnyn2 x2n + yn2 xn+1 = x2 + y2 and yn+1 = x + y .
nn nn
(a) Prove that xn+1 − yn+1 = −(x3n − yn3 )(xn − yn) for all
positive integers n.
(xn +yn)(x2n +yn2) (b) Show that 0 < xn ≤ yn for all positive
integers n.
Hence, prove...
Problem 1 Let {ak} and {bk} be sequences of positive real numbers. Assume that lim “k...
Problem 1 Let {ak} and {bk} be sequences of positive real numbers. Assume that lim “k = 0. k+oo bk 1. Prove that if ) bk converges, so does 'ak k=1 k=1 2. If ) bk diverges, is it necessary that ) ak diverges? k=1 k=1
Can you solve No.6 6. Let (a.)and b) be bounded sequences in R .a. Prove that lima. +İimb, siim (a, + b.) s ima, + İim br b. Prove that lim (-a)lima . Given an example to show that equality need n...
Can you solve No.6
6. Let (a.)and b) be bounded sequences in R .a. Prove that lima. +İimb, siim (a, + b.) s ima, + İim br b. Prove that lim (-a)lima . Given an example to show that equality need not hold in (a) If o, and b, are positive for all n, prove that lim (a)s(im a)mb). provided the product on the right is 7. not of the form 0 oo. b. Need equality hold in (a)?
6....
18. If ai, az, as,... is a bounded sequence of real numbers, define lim sup an (also denoted lim an) to be --+ n+ l.u.b. {z ER: an > & for an infinite number of integers n} and define lim inf an (also denoted lim an) to be g.l.b. {ER: An <for an infinite number of integers n}. Prove that lim inf an Slim sup an, with the equality holding if and only if the sequence converges. 19. Let ai,...
can you please explain a and b thanks
Fourier Analysis
See are two finite sequences of complex numben 7. Suppose (an)- and (bn)1 Let Br= bn denote the partial sums of the series b with the conventicn 1 Bo=0. (a) Prove the summation by parts formula N-1 anbn aNBN- aM BM-1 (an+1-an)B n M n-M (b) Deduce from this formula Dirichlet's test for convergence of a series: if the partial sums of the seriesb are bounded, and fan} is a...
2.13.4 2.13.5 Show that lim supno (-X) = -(liminf ,-Xn). If two sequences {an) and {bn} satisfy the inequality an <b, for all sufficiently large n, show that limsupan Slim sup bn and liminfa, <liminf bn. 100 2.13.6 Show that lim, 100 Xn = o if and only if lim sup.Xn = liminf xn = c. n-00 2.13.7 Show that if lim sup a n = L for a finite real number L and € > 0, then an >...
{x_n} and {y_n} are sequences of positive real numbers
AC fn→oo > O, prove tha m in yn lim xn 0 implies lim yn_0
Let (xn) be a bounded sequence
of real numbers, and put u = lim supn→∞ xn . Let E be the set
consisting of the limits of all convergent subsequences of (xn).
Show that u ∈ E and that u = sup(E).
Formulate and prove a similar result for lim infn→∞ xn .
Thank you!
7. Let (Fm) be a bounded sequence of real numbers, and put u-lim supn→oorn . Let E be the set consisting of the limits of...
Let {an} m-o and {bn}ņ=be any two sequences of real numbers, we define the following: N • For any real number L € R, we write an = L if and only if lim Lan = L. N-0 n=0 n=0 X • We write an = bn if and only if there is a real number L such that n=0 n=0 I and Σ. = L. Select all the correct sentences in the following list: X η (Α) Σ Σ...
Prove the following
Definition 6.6.1 (Subsequences). Let (an) =) and (bn), m=0 be sequences of real numbers. We say that (bn)is a subsequence of (an) a=iff there exists a function f :N + N which is strictly increasing (i.e., f(n + 1) > f(n) for all n EN) such that bn = f(n) for all n E N.
5. Let {xn} and {yn} be sequences of real numbers such that x1 =
2 and y1 = 8 and for n = 1,2,3,···
x2nyn + xnyn2 x2n + yn2 xn+1 = x2 + y2 and yn+1 = x + y .
nn nn
(a) Prove that xn+1 − yn+1 = −(x3n − yn3 )(xn − yn) for all
positive integers n.
(xn +yn)(x2n +yn2) (b) Show that 0 < xn ≤ yn for all positive
integers n.
Hence, prove...
Problem 1 Let {ak} and {bk} be sequences of positive real numbers. Assume that lim “k = 0. k+oo bk 1. Prove that if ) bk converges, so does 'ak k=1 k=1 2. If ) bk diverges, is it necessary that ) ak diverges? k=1 k=1
Can you solve No.6
6. Let (a.)and b) be bounded sequences in R .a. Prove that lima. +İimb, siim (a, + b.) s ima, + İim br b. Prove that lim (-a)lima . Given an example to show that equality need not hold in (a) If o, and b, are positive for all n, prove that lim (a)s(im a)mb). provided the product on the right is 7. not of the form 0 oo. b. Need equality hold in (a)?
6....
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