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find $limsup$ and $liminf$ of $a_n={sqrt{n} - lfloorsqrt{n}rfloor$ $ninmathbb{N}}$ [duplicate]
This question already has an answer here:
Supremum of $sqrt{n} -leftlfloorsqrt{n}rightrfloor $
3 answers
I have to find $limsup$ and $liminf$ of $a_n={sqrt{n} - lfloorsqrt{n}rfloor : n inmathbb{N}}$
I suppose that I have to relate to subsequences .
First one is $a_{n_k}$ for all n that maintains $sqrt ninmathbb {N}$ , so the limit is $0$.
Second one $a_{n_j}$ includes the rest of the elements.I suppose it converges to 1, but dont know how to prove it.
sequences-and-series limits limsup-and-liminf
edited Nov 21 '18 at 9:08
Chinnapparaj R
5,2951826
asked Nov 21 '18 at 8:58
Alex ˢᵅˢʰᵅ Druzina
144
marked as duplicate by Martin R, Chinnapparaj R, Community♦ Nov 21 '18 at 9:23
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
add a comment |
This question already has an answer here:
Supremum of $sqrt{n} -leftlfloorsqrt{n}rightrfloor $
3 answers
I have to find $limsup$ and $liminf$ of $a_n={sqrt{n} - lfloorsqrt{n}rfloor : n inmathbb{N}}$
I suppose that I have to relate to subsequences .
First one is $a_{n_k}$ for all n that maintains $sqrt ninmathbb {N}$ , so the limit is $0$.
Second one $a_{n_j}$ includes the rest of the elements.I suppose it converges to 1, but dont know how to prove it.
sequences-and-series limits limsup-and-liminf
edited Nov 21 '18 at 9:08
Chinnapparaj R
5,2951826
asked Nov 21 '18 at 8:58
Alex ˢᵅˢʰᵅ Druzina
144
marked as duplicate by Martin R, Chinnapparaj R, Community♦ Nov 21 '18 at 9:23
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
What's a subseries?
– José Carlos Santos
Nov 21 '18 at 9:00
@JoséCarlosSantos I guess subsequence
– Faustus
Nov 21 '18 at 9:00
Yes, thank you. My english is no so great, espetially in math terminilogy
– Alex ˢᵅˢʰᵅ Druzina
Nov 21 '18 at 9:02
I dont understand lim sup or lim inf.
– Love Invariants
Nov 21 '18 at 9:07
Gimusi.Have a look at the link where Yves answers this question, in particular his shorter answer.Vow!:)Greetings.
– Peter Szilas
Nov 21 '18 at 9:26
add a comment |
This question already has an answer here:
Supremum of $sqrt{n} -leftlfloorsqrt{n}rightrfloor $
3 answers
I have to find $limsup$ and $liminf$ of $a_n={sqrt{n} - lfloorsqrt{n}rfloor : n inmathbb{N}}$
I suppose that I have to relate to subsequences .
First one is $a_{n_k}$ for all n that maintains $sqrt ninmathbb {N}$ , so the limit is $0$.
Second one $a_{n_j}$ includes the rest of the elements.I suppose it converges to 1, but dont know how to prove it.
sequences-and-series limits limsup-and-liminf
edited Nov 21 '18 at 9:08
Chinnapparaj R
5,2951826
asked Nov 21 '18 at 8:58
Alex ˢᵅˢʰᵅ Druzina
144
This question already has an answer here:
Supremum of $sqrt{n} -leftlfloorsqrt{n}rightrfloor $
3 answers
I have to find $limsup$ and $liminf$ of $a_n={sqrt{n} - lfloorsqrt{n}rfloor : n inmathbb{N}}$
I suppose that I have to relate to subsequences .
First one is $a_{n_k}$ for all n that maintains $sqrt ninmathbb {N}$ , so the limit is $0$.
Second one $a_{n_j}$ includes the rest of the elements.I suppose it converges to 1, but dont know how to prove it.
This question already has an answer here:
Supremum of $sqrt{n} -leftlfloorsqrt{n}rightrfloor $
3 answers
sequences-and-series limits limsup-and-liminf
sequences-and-series limits limsup-and-liminf
edited Nov 21 '18 at 9:08
Chinnapparaj R
5,2951826
asked Nov 21 '18 at 8:58
Alex ˢᵅˢʰᵅ Druzina
144
edited Nov 21 '18 at 9:08
Chinnapparaj R
5,2951826
asked Nov 21 '18 at 8:58
Alex ˢᵅˢʰᵅ Druzina
144
edited Nov 21 '18 at 9:08
Chinnapparaj R
5,2951826
edited Nov 21 '18 at 9:08
Chinnapparaj R
5,2951826
edited Nov 21 '18 at 9:08
Chinnapparaj R
5,2951826
5,2951826
asked Nov 21 '18 at 8:58
Alex ˢᵅˢʰᵅ Druzina
144
asked Nov 21 '18 at 8:58
Alex ˢᵅˢʰᵅ Druzina
144
asked Nov 21 '18 at 8:58
Alex ˢᵅˢʰᵅ Druzina
144
144
marked as duplicate by Martin R, Chinnapparaj R, Community♦ Nov 21 '18 at 9:23
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by Martin R, Chinnapparaj R, Community♦ Nov 21 '18 at 9:23
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
What's a subseries?
– José Carlos Santos
Nov 21 '18 at 9:00
@JoséCarlosSantos I guess subsequence
– Faustus
Nov 21 '18 at 9:00
Yes, thank you. My english is no so great, espetially in math terminilogy
– Alex ˢᵅˢʰᵅ Druzina
Nov 21 '18 at 9:02
I dont understand lim sup or lim inf.
– Love Invariants
Nov 21 '18 at 9:07
Gimusi.Have a look at the link where Yves answers this question, in particular his shorter answer.Vow!:)Greetings.
– Peter Szilas
Nov 21 '18 at 9:26
add a comment |
What's a subseries?
– José Carlos Santos
Nov 21 '18 at 9:00
@JoséCarlosSantos I guess subsequence
– Faustus
Nov 21 '18 at 9:00
Yes, thank you. My english is no so great, espetially in math terminilogy
– Alex ˢᵅˢʰᵅ Druzina
Nov 21 '18 at 9:02
I dont understand lim sup or lim inf.
– Love Invariants
Nov 21 '18 at 9:07
Gimusi.Have a look at the link where Yves answers this question, in particular his shorter answer.Vow!:)Greetings.
– Peter Szilas
Nov 21 '18 at 9:26
What's a subseries?
– José Carlos Santos
Nov 21 '18 at 9:00
What's a subseries?
– José Carlos Santos
Nov 21 '18 at 9:00
@JoséCarlosSantos I guess subsequence
– Faustus
Nov 21 '18 at 9:00
@JoséCarlosSantos I guess subsequence
– Faustus
Nov 21 '18 at 9:00
Yes, thank you. My english is no so great, espetially in math terminilogy
– Alex ˢᵅˢʰᵅ Druzina
Nov 21 '18 at 9:02
Yes, thank you. My english is no so great, espetially in math terminilogy
– Alex ˢᵅˢʰᵅ Druzina
Nov 21 '18 at 9:02
I dont understand lim sup or lim inf.
– Love Invariants
Nov 21 '18 at 9:07
I dont understand lim sup or lim inf.
– Love Invariants
Nov 21 '18 at 9:07
Gimusi.Have a look at the link where Yves answers this question, in particular his shorter answer.Vow!:)Greetings.
– Peter Szilas
Nov 21 '18 at 9:26
Gimusi.Have a look at the link where Yves answers this question, in particular his shorter answer.Vow!:)Greetings.
– Peter Szilas
Nov 21 '18 at 9:26
add a comment |
1 Answer
1
active
oldest
votes
We have that
$$a_n={sqrt{n} - lfloorsqrt{n}rfloor ge 0$$
and for $n=m^2 implies a_n=0$ then $liminf a_n =0$.
For $limsup$ we have
$$a_n={sqrt{n} - lfloorsqrt{n}rfloor le 1$$
and for any $epsilon>0$ we can find $n=m^2-1$ such that $sqrt{n} - lfloorsqrt{n}rfloor=1-epsilon$.
answered Nov 21 '18 at 9:01
gimusi
1
That is exacly what Im asking. I have already found lim inf but, has no idea about lim sup
– Alex ˢᵅˢʰᵅ Druzina
Nov 21 '18 at 9:04
You might notice that a possible duplicate target has been pointed out in the meantime. Your approach is essentially the same as math.stackexchange.com/a/2015624.
– Martin R
Nov 21 '18 at 9:12
@MartinR Thanks, I see that now!
– gimusi
Nov 21 '18 at 9:13
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
We have that
$$a_n={sqrt{n} - lfloorsqrt{n}rfloor ge 0$$
and for $n=m^2 implies a_n=0$ then $liminf a_n =0$.
For $limsup$ we have
$$a_n={sqrt{n} - lfloorsqrt{n}rfloor le 1$$
and for any $epsilon>0$ we can find $n=m^2-1$ such that $sqrt{n} - lfloorsqrt{n}rfloor=1-epsilon$.
answered Nov 21 '18 at 9:01
gimusi
1
That is exacly what Im asking. I have already found lim inf but, has no idea about lim sup
– Alex ˢᵅˢʰᵅ Druzina
Nov 21 '18 at 9:04
You might notice that a possible duplicate target has been pointed out in the meantime. Your approach is essentially the same as math.stackexchange.com/a/2015624.
– Martin R
Nov 21 '18 at 9:12
@MartinR Thanks, I see that now!
– gimusi
Nov 21 '18 at 9:13
add a comment |
We have that
$$a_n={sqrt{n} - lfloorsqrt{n}rfloor ge 0$$
and for $n=m^2 implies a_n=0$ then $liminf a_n =0$.
For $limsup$ we have
$$a_n={sqrt{n} - lfloorsqrt{n}rfloor le 1$$
and for any $epsilon>0$ we can find $n=m^2-1$ such that $sqrt{n} - lfloorsqrt{n}rfloor=1-epsilon$.
answered Nov 21 '18 at 9:01
gimusi
1
That is exacly what Im asking. I have already found lim inf but, has no idea about lim sup
– Alex ˢᵅˢʰᵅ Druzina
Nov 21 '18 at 9:04
You might notice that a possible duplicate target has been pointed out in the meantime. Your approach is essentially the same as math.stackexchange.com/a/2015624.
– Martin R
Nov 21 '18 at 9:12
@MartinR Thanks, I see that now!
– gimusi
Nov 21 '18 at 9:13
add a comment |
We have that
$$a_n={sqrt{n} - lfloorsqrt{n}rfloor ge 0$$
and for $n=m^2 implies a_n=0$ then $liminf a_n =0$.
For $limsup$ we have
$$a_n={sqrt{n} - lfloorsqrt{n}rfloor le 1$$
and for any $epsilon>0$ we can find $n=m^2-1$ such that $sqrt{n} - lfloorsqrt{n}rfloor=1-epsilon$.
answered Nov 21 '18 at 9:01
gimusi
1
We have that
$$a_n={sqrt{n} - lfloorsqrt{n}rfloor ge 0$$
and for $n=m^2 implies a_n=0$ then $liminf a_n =0$.
For $limsup$ we have
$$a_n={sqrt{n} - lfloorsqrt{n}rfloor le 1$$
and for any $epsilon>0$ we can find $n=m^2-1$ such that $sqrt{n} - lfloorsqrt{n}rfloor=1-epsilon$.
answered Nov 21 '18 at 9:01
gimusi
1
edited Nov 21 '18 at 9:11
answered Nov 21 '18 at 9:01
gimusi
1
answered Nov 21 '18 at 9:01
gimusi
1
answered Nov 21 '18 at 9:01
gimusi
1
1
That is exacly what Im asking. I have already found lim inf but, has no idea about lim sup
– Alex ˢᵅˢʰᵅ Druzina
Nov 21 '18 at 9:04
You might notice that a possible duplicate target has been pointed out in the meantime. Your approach is essentially the same as math.stackexchange.com/a/2015624.
– Martin R
Nov 21 '18 at 9:12
@MartinR Thanks, I see that now!
– gimusi
Nov 21 '18 at 9:13
add a comment |
That is exacly what Im asking. I have already found lim inf but, has no idea about lim sup
– Alex ˢᵅˢʰᵅ Druzina
Nov 21 '18 at 9:04
You might notice that a possible duplicate target has been pointed out in the meantime. Your approach is essentially the same as math.stackexchange.com/a/2015624.
– Martin R
Nov 21 '18 at 9:12
@MartinR Thanks, I see that now!
– gimusi
Nov 21 '18 at 9:13
That is exacly what Im asking. I have already found lim inf but, has no idea about lim sup
– Alex ˢᵅˢʰᵅ Druzina
Nov 21 '18 at 9:04
That is exacly what Im asking. I have already found lim inf but, has no idea about lim sup
– Alex ˢᵅˢʰᵅ Druzina
Nov 21 '18 at 9:04
You might notice that a possible duplicate target has been pointed out in the meantime. Your approach is essentially the same as math.stackexchange.com/a/2015624.
– Martin R
Nov 21 '18 at 9:12
You might notice that a possible duplicate target has been pointed out in the meantime. Your approach is essentially the same as math.stackexchange.com/a/2015624.
– Martin R
Nov 21 '18 at 9:12
@MartinR Thanks, I see that now!
– gimusi
Nov 21 '18 at 9:13
@MartinR Thanks, I see that now!
– gimusi
Nov 21 '18 at 9:13
add a comment |
What's a subseries?
– José Carlos Santos
Nov 21 '18 at 9:00
@JoséCarlosSantos I guess subsequence
– Faustus
Nov 21 '18 at 9:00
Yes, thank you. My english is no so great, espetially in math terminilogy
– Alex ˢᵅˢʰᵅ Druzina
Nov 21 '18 at 9:02
I dont understand lim sup or lim inf.
– Love Invariants
Nov 21 '18 at 9:07
Gimusi.Have a look at the link where Yves answers this question, in particular his shorter answer.Vow!:)Greetings.
– Peter Szilas
Nov 21 '18 at 9:26