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et (x) be a bounded sequence with positive terms. Show that lim(n2/x,) = +0 et (x)...
For each statement: True or false? Explain? If the terms sn of a convergent sequence are all positive then lim sn is positive. If the sequence sn of positive terms is unbounded, then the sequence has...
For each statement: True or false? Explain?
If the terms sn of a convergent sequence are all positive then lim sn is positive. If the sequence sn of positive terms is unbounded, then the sequence has a term greater than a million. If the sequence sn of positive terms is unbounded, then the sequence has an infinite number of terms greater than a million. If a sequence sn is convergent, then the terms sn tend to zero as n increases....
6. Suppose that {x,] is a sequence of positive numbers and limA = a Show that...
6. Suppose that {x,] is a sequence of positive numbers and limA = a Show that if L> 1 then lim x =00, and if L < 1 lim x = 0 n+02 b. Construct a sequence of positive numbers {x,} such that lim * = 1 and the sequence {x} diverges. c. Let k E N and a > 1 Show that lim = 0. O LIVE
Exercise 2.3.9. (a) Let (an) be a bounded (not necessarily convergent) sequence, and assume lim bn...
Exercise 2.3.9. (a) Let (an) be a bounded (not necessarily convergent) sequence, and assume lim bn = 0. Show that lim(anon) = 0. Why are we not allowed to use the Algebraic Limit Theorem to prove this?
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,...
***You must follow the comments*** Topic: Mathematical Real Analysis - Let (xn) be a bounded sequence...
***You must
follow the comments***
Topic: Mathematical
Real Analysis
- Let (xn) be a
bounded sequence ((xn) is not necessarily convergent), and assume
that yn → 0. Show that lim n→∞ (xnyn) = 0.
Question1.
All the
solution state that there exists M >0 and xn<=M . My question
is that why M always be bigger than 0 and Why it is bounded above ?
why it is not m<=xn bounded below????
Question.
2.
if the
sequence is convergent, then...
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...
Q7 You are given the sequence n2 In(x)= n2 + of functions on the domain [0,...
Q7 You are given the sequence n2 In(x)= n2 + of functions on the domain [0, oo) where 0 < a < 2. Determine the range of o for which the sequence is uniformly convergent.
Example: Let {xn} be a sequence of real numbers. Show that Proposition 0.1 1. If r...
Example: Let {xn} be a sequence of real numbers. Show that Proposition 0.1 1. If r is bounded above, x = lim sup (r) if and only if For all 0 there is an NEN, such that x <x+e whenevern > N, and b. For all >0 and all M, there is n > M with x - e< In a.
Example: Let {xn} be a sequence of real numbers. Show that Proposition 0.1 1. If r is bounded above,...
13 14 Exercise 13: Let (xn) be a bounded sequence a S be the set of...
13 14
Exercise 13: Let (xn) be a bounded sequence a S be the set of limit points of (n), i.e. S:{xER there exists a subsequence () s.t. lim } ko0 Show lim inf inf S n-o0 Hint: See lecture for proof lim sup Exercise 14: (Caesaro revisited) Let (x) be a convergent sequence. Let (yn) be the sequence given by Yn= n for all n E N. Show that lim sup y lim sup n n-+00 n o0
x-4 a) lim x-2 X-2 xe* b) lim *-01-et lim (1+x)'* X+00
x-4 a) lim x-2 X-2 xe* b) lim *-01-et lim (1+x)'* X+00
For each statement: True or false? Explain?
If the terms sn of a convergent sequence are all positive then lim sn is positive. If the sequence sn of positive terms is unbounded, then the sequence has a term greater than a million. If the sequence sn of positive terms is unbounded, then the sequence has an infinite number of terms greater than a million. If a sequence sn is convergent, then the terms sn tend to zero as n increases....
6. Suppose that {x,] is a sequence of positive numbers and limA = a Show that if L> 1 then lim x =00, and if L < 1 lim x = 0 n+02 b. Construct a sequence of positive numbers {x,} such that lim * = 1 and the sequence {x} diverges. c. Let k E N and a > 1 Show that lim = 0. O LIVE
Exercise 2.3.9. (a) Let (an) be a bounded (not necessarily convergent) sequence, and assume lim bn = 0. Show that lim(anon) = 0. Why are we not allowed to use the Algebraic Limit Theorem to prove this?
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,...
***You must
follow the comments***
Topic: Mathematical
Real Analysis
- Let (xn) be a
bounded sequence ((xn) is not necessarily convergent), and assume
that yn → 0. Show that lim n→∞ (xnyn) = 0.
Question1.
All the
solution state that there exists M >0 and xn<=M . My question
is that why M always be bigger than 0 and Why it is bounded above ?
why it is not m<=xn bounded below????
Question.
2.
if the
sequence is convergent, then...
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...
Q7 You are given the sequence n2 In(x)= n2 + of functions on the domain [0, oo) where 0 < a < 2. Determine the range of o for which the sequence is uniformly convergent.
Example: Let {xn} be a sequence of real numbers. Show that Proposition 0.1 1. If r is bounded above, x = lim sup (r) if and only if For all 0 there is an NEN, such that x <x+e whenevern > N, and b. For all >0 and all M, there is n > M with x - e< In a.
Example: Let {xn} be a sequence of real numbers. Show that Proposition 0.1 1. If r is bounded above,...
13 14
Exercise 13: Let (xn) be a bounded sequence a S be the set of limit points of (n), i.e. S:{xER there exists a subsequence () s.t. lim } ko0 Show lim inf inf S n-o0 Hint: See lecture for proof lim sup Exercise 14: (Caesaro revisited) Let (x) be a convergent sequence. Let (yn) be the sequence given by Yn= n for all n E N. Show that lim sup y lim sup n n-+00 n o0
x-4 a) lim x-2 X-2 xe* b) lim *-01-et lim (1+x)'* X+00
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