If $(a_n)$ diverges, then when does it tend to infinity
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Let $(a_n)$ be a sequence of real numbers. Consider the two points,
- 1.$quad$ $(a_n)$ diverges,
- 2.$quad$ $(a_n)$ tends to plus infinity.
Using the fact that an unbounded sequence diverges, the second point implies the first one. What conditions should $(a_n)$ have to make first point to imply the second one? It seems obvious to me that $(a_n)$ has to be non-negative eventually or increasing eventually, but I do not know how to prove it.
Example: Let $(b_n)$ be a sequence of positive numbers, and define $s_N=sum_n=1^Nb_n$. If $sum_n=1^inftyb_n$ diverges, then it must be the case that $s_Ntoinfty$ as $Ntoinfty$, which is reasonable. But I want to know the proof. This is why I asked a question.
real-analysis convergence
asked Aug 30 at 9:33
UnknownW
917821
 |Â
show 3 more comments
up vote
0
down vote
favorite
Let $(a_n)$ be a sequence of real numbers. Consider the two points,
- 1.$quad$ $(a_n)$ diverges,
- 2.$quad$ $(a_n)$ tends to plus infinity.
Using the fact that an unbounded sequence diverges, the second point implies the first one. What conditions should $(a_n)$ have to make first point to imply the second one? It seems obvious to me that $(a_n)$ has to be non-negative eventually or increasing eventually, but I do not know how to prove it.
Example: Let $(b_n)$ be a sequence of positive numbers, and define $s_N=sum_n=1^Nb_n$. If $sum_n=1^inftyb_n$ diverges, then it must be the case that $s_Ntoinfty$ as $Ntoinfty$, which is reasonable. But I want to know the proof. This is why I asked a question.
real-analysis convergence
asked Aug 30 at 9:33
UnknownW
917821
3
I don't think there's really a way to restate this better than the definition of tending to infinity.
– Ian
Aug 30 at 9:35
For every $R>0$, you should be able to find an $N geq 1$ so that $a_n geq R$ whenever $n geq N$.
– Sobi
Aug 30 at 9:36
Now that you edited point 2 with PLUS infinity: it is often stated in books that monotone divergent sequences tend either to plus or minus infinty. In this case you would just need $a_n$ to be non decreasing
– Nicolò
Aug 30 at 9:42
$lim inf a_n = infty$, but that is just paraphrasing point 2...
– nicomezi
Aug 30 at 9:42
1
"has to be non-negative eventually" Yes. "or increasing eventually" No. Counterexample: $a_n=n+(-1)^n$.
– Did
Aug 30 at 9:57
 |Â
show 3 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $(a_n)$ be a sequence of real numbers. Consider the two points,
- 1.$quad$ $(a_n)$ diverges,
- 2.$quad$ $(a_n)$ tends to plus infinity.
Using the fact that an unbounded sequence diverges, the second point implies the first one. What conditions should $(a_n)$ have to make first point to imply the second one? It seems obvious to me that $(a_n)$ has to be non-negative eventually or increasing eventually, but I do not know how to prove it.
Example: Let $(b_n)$ be a sequence of positive numbers, and define $s_N=sum_n=1^Nb_n$. If $sum_n=1^inftyb_n$ diverges, then it must be the case that $s_Ntoinfty$ as $Ntoinfty$, which is reasonable. But I want to know the proof. This is why I asked a question.
real-analysis convergence
asked Aug 30 at 9:33
UnknownW
917821
Let $(a_n)$ be a sequence of real numbers. Consider the two points,
- 1.$quad$ $(a_n)$ diverges,
- 2.$quad$ $(a_n)$ tends to plus infinity.
Using the fact that an unbounded sequence diverges, the second point implies the first one. What conditions should $(a_n)$ have to make first point to imply the second one? It seems obvious to me that $(a_n)$ has to be non-negative eventually or increasing eventually, but I do not know how to prove it.
Example: Let $(b_n)$ be a sequence of positive numbers, and define $s_N=sum_n=1^Nb_n$. If $sum_n=1^inftyb_n$ diverges, then it must be the case that $s_Ntoinfty$ as $Ntoinfty$, which is reasonable. But I want to know the proof. This is why I asked a question.
real-analysis convergence
real-analysis convergence
asked Aug 30 at 9:33
UnknownW
917821
asked Aug 30 at 9:33
UnknownW
917821
edited Aug 30 at 9:43
asked Aug 30 at 9:33
UnknownW
917821
asked Aug 30 at 9:33
UnknownW
917821
asked Aug 30 at 9:33
UnknownW
917821
917821
3
I don't think there's really a way to restate this better than the definition of tending to infinity.
– Ian
Aug 30 at 9:35
For every $R>0$, you should be able to find an $N geq 1$ so that $a_n geq R$ whenever $n geq N$.
– Sobi
Aug 30 at 9:36
Now that you edited point 2 with PLUS infinity: it is often stated in books that monotone divergent sequences tend either to plus or minus infinty. In this case you would just need $a_n$ to be non decreasing
– Nicolò
Aug 30 at 9:42
$lim inf a_n = infty$, but that is just paraphrasing point 2...
– nicomezi
Aug 30 at 9:42
1
"has to be non-negative eventually" Yes. "or increasing eventually" No. Counterexample: $a_n=n+(-1)^n$.
– Did
Aug 30 at 9:57
 |Â
show 3 more comments
3
I don't think there's really a way to restate this better than the definition of tending to infinity.
– Ian
Aug 30 at 9:35
For every $R>0$, you should be able to find an $N geq 1$ so that $a_n geq R$ whenever $n geq N$.
– Sobi
Aug 30 at 9:36
Now that you edited point 2 with PLUS infinity: it is often stated in books that monotone divergent sequences tend either to plus or minus infinty. In this case you would just need $a_n$ to be non decreasing
– Nicolò
Aug 30 at 9:42
$lim inf a_n = infty$, but that is just paraphrasing point 2...
– nicomezi
Aug 30 at 9:42
1
"has to be non-negative eventually" Yes. "or increasing eventually" No. Counterexample: $a_n=n+(-1)^n$.
– Did
Aug 30 at 9:57
3
3
I don't think there's really a way to restate this better than the definition of tending to infinity.
– Ian
Aug 30 at 9:35
I don't think there's really a way to restate this better than the definition of tending to infinity.
– Ian
Aug 30 at 9:35
For every $R>0$, you should be able to find an $N geq 1$ so that $a_n geq R$ whenever $n geq N$.
– Sobi
Aug 30 at 9:36
For every $R>0$, you should be able to find an $N geq 1$ so that $a_n geq R$ whenever $n geq N$.
– Sobi
Aug 30 at 9:36
Now that you edited point 2 with PLUS infinity: it is often stated in books that monotone divergent sequences tend either to plus or minus infinty. In this case you would just need $a_n$ to be non decreasing
– Nicolò
Aug 30 at 9:42
Now that you edited point 2 with PLUS infinity: it is often stated in books that monotone divergent sequences tend either to plus or minus infinty. In this case you would just need $a_n$ to be non decreasing
– Nicolò
Aug 30 at 9:42
$lim inf a_n = infty$, but that is just paraphrasing point 2...
– nicomezi
Aug 30 at 9:42
$lim inf a_n = infty$, but that is just paraphrasing point 2...
– nicomezi
Aug 30 at 9:42
1
1
"has to be non-negative eventually" Yes. "or increasing eventually" No. Counterexample: $a_n=n+(-1)^n$.
– Did
Aug 30 at 9:57
"has to be non-negative eventually" Yes. "or increasing eventually" No. Counterexample: $a_n=n+(-1)^n$.
– Did
Aug 30 at 9:57
 |Â
show 3 more comments
2 Answers
2
active
oldest
votes
up vote
1
down vote
If $(a_n)$ is (weakly) increasing, then by the monotone sequence theorem it either converges to a finite value or diverges to $+infty$. However, there are sequences that tend to $+infty$ that are not eventually increasing, for example
$(a_n)=(1,2,3,2,3,4,3,4,5,4,5,6,5,6,7,6,7,8,...)$.
answered Aug 30 at 9:51
Kusma
3,355218
1
To the proposer: Another example: $a_2n-1=n$ and $a_2n=2^n.$
– DanielWainfleet
Aug 30 at 14:31
add a comment |Â
up vote
0
down vote
"Eventually non-negative" is necessary but not sufficient.
"Eventually increasing" is not necessary and not sufficient.
"Unbounded" is necessary but not sufficient.
"Eventually increasing and unbounded" is not necessary but sufficient.
As said in comments, "larger than any number $u$ as of some $n(u)$" is the straightest requirement. This can also be phrased as "bounded below by an increasing unbounded function". (Intuitively, closer and closer to infinity.)
answered Aug 30 at 10:11
Yves Daoust
114k665209
What does "necessary" and "sufficient" mean in terms of mathematical symbols? For example, does >"Eventually non-negative" is necessary but not sufficient< mean that "if $(a_n)$ is eventually non-negative and diverges, then $(a_n)$ tends to plus infinity" but "if $(a_n)$ is eventually non-negative and tends to plus infinity, then it does not diverge"?
– UnknownW
Aug 30 at 20:28
@UnknownW: en.wikipedia.org/wiki/Necessity_and_sufficiency
– Yves Daoust
Aug 30 at 21:25
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
If $(a_n)$ is (weakly) increasing, then by the monotone sequence theorem it either converges to a finite value or diverges to $+infty$. However, there are sequences that tend to $+infty$ that are not eventually increasing, for example
$(a_n)=(1,2,3,2,3,4,3,4,5,4,5,6,5,6,7,6,7,8,...)$.
answered Aug 30 at 9:51
Kusma
3,355218
1
To the proposer: Another example: $a_2n-1=n$ and $a_2n=2^n.$
– DanielWainfleet
Aug 30 at 14:31
add a comment |Â
up vote
1
down vote
If $(a_n)$ is (weakly) increasing, then by the monotone sequence theorem it either converges to a finite value or diverges to $+infty$. However, there are sequences that tend to $+infty$ that are not eventually increasing, for example
$(a_n)=(1,2,3,2,3,4,3,4,5,4,5,6,5,6,7,6,7,8,...)$.
answered Aug 30 at 9:51
Kusma
3,355218
1
To the proposer: Another example: $a_2n-1=n$ and $a_2n=2^n.$
– DanielWainfleet
Aug 30 at 14:31
add a comment |Â
up vote
1
down vote
up vote
1
down vote
If $(a_n)$ is (weakly) increasing, then by the monotone sequence theorem it either converges to a finite value or diverges to $+infty$. However, there are sequences that tend to $+infty$ that are not eventually increasing, for example
$(a_n)=(1,2,3,2,3,4,3,4,5,4,5,6,5,6,7,6,7,8,...)$.
answered Aug 30 at 9:51
Kusma
3,355218
If $(a_n)$ is (weakly) increasing, then by the monotone sequence theorem it either converges to a finite value or diverges to $+infty$. However, there are sequences that tend to $+infty$ that are not eventually increasing, for example
$(a_n)=(1,2,3,2,3,4,3,4,5,4,5,6,5,6,7,6,7,8,...)$.
answered Aug 30 at 9:51
Kusma
3,355218
answered Aug 30 at 9:51
Kusma
3,355218
answered Aug 30 at 9:51
Kusma
3,355218
answered Aug 30 at 9:51
Kusma
3,355218
3,355218
1
To the proposer: Another example: $a_2n-1=n$ and $a_2n=2^n.$
– DanielWainfleet
Aug 30 at 14:31
add a comment |Â
1
To the proposer: Another example: $a_2n-1=n$ and $a_2n=2^n.$
– DanielWainfleet
Aug 30 at 14:31
1
1
To the proposer: Another example: $a_2n-1=n$ and $a_2n=2^n.$
– DanielWainfleet
Aug 30 at 14:31
To the proposer: Another example: $a_2n-1=n$ and $a_2n=2^n.$
– DanielWainfleet
Aug 30 at 14:31
add a comment |Â
up vote
0
down vote
"Eventually non-negative" is necessary but not sufficient.
"Eventually increasing" is not necessary and not sufficient.
"Unbounded" is necessary but not sufficient.
"Eventually increasing and unbounded" is not necessary but sufficient.
As said in comments, "larger than any number $u$ as of some $n(u)$" is the straightest requirement. This can also be phrased as "bounded below by an increasing unbounded function". (Intuitively, closer and closer to infinity.)
answered Aug 30 at 10:11
Yves Daoust
114k665209
What does "necessary" and "sufficient" mean in terms of mathematical symbols? For example, does >"Eventually non-negative" is necessary but not sufficient< mean that "if $(a_n)$ is eventually non-negative and diverges, then $(a_n)$ tends to plus infinity" but "if $(a_n)$ is eventually non-negative and tends to plus infinity, then it does not diverge"?
– UnknownW
Aug 30 at 20:28
@UnknownW: en.wikipedia.org/wiki/Necessity_and_sufficiency
– Yves Daoust
Aug 30 at 21:25
add a comment |Â
up vote
0
down vote
"Eventually non-negative" is necessary but not sufficient.
"Eventually increasing" is not necessary and not sufficient.
"Unbounded" is necessary but not sufficient.
"Eventually increasing and unbounded" is not necessary but sufficient.
As said in comments, "larger than any number $u$ as of some $n(u)$" is the straightest requirement. This can also be phrased as "bounded below by an increasing unbounded function". (Intuitively, closer and closer to infinity.)
answered Aug 30 at 10:11
Yves Daoust
114k665209
What does "necessary" and "sufficient" mean in terms of mathematical symbols? For example, does >"Eventually non-negative" is necessary but not sufficient< mean that "if $(a_n)$ is eventually non-negative and diverges, then $(a_n)$ tends to plus infinity" but "if $(a_n)$ is eventually non-negative and tends to plus infinity, then it does not diverge"?
– UnknownW
Aug 30 at 20:28
@UnknownW: en.wikipedia.org/wiki/Necessity_and_sufficiency
– Yves Daoust
Aug 30 at 21:25
add a comment |Â
up vote
0
down vote
up vote
0
down vote
"Eventually non-negative" is necessary but not sufficient.
"Eventually increasing" is not necessary and not sufficient.
"Unbounded" is necessary but not sufficient.
"Eventually increasing and unbounded" is not necessary but sufficient.
As said in comments, "larger than any number $u$ as of some $n(u)$" is the straightest requirement. This can also be phrased as "bounded below by an increasing unbounded function". (Intuitively, closer and closer to infinity.)
answered Aug 30 at 10:11
Yves Daoust
114k665209
"Eventually non-negative" is necessary but not sufficient.
"Eventually increasing" is not necessary and not sufficient.
"Unbounded" is necessary but not sufficient.
"Eventually increasing and unbounded" is not necessary but sufficient.
As said in comments, "larger than any number $u$ as of some $n(u)$" is the straightest requirement. This can also be phrased as "bounded below by an increasing unbounded function". (Intuitively, closer and closer to infinity.)
answered Aug 30 at 10:11
Yves Daoust
114k665209
edited Aug 30 at 10:18
answered Aug 30 at 10:11
Yves Daoust
114k665209
answered Aug 30 at 10:11
Yves Daoust
114k665209
answered Aug 30 at 10:11
Yves Daoust
114k665209
114k665209
What does "necessary" and "sufficient" mean in terms of mathematical symbols? For example, does >"Eventually non-negative" is necessary but not sufficient< mean that "if $(a_n)$ is eventually non-negative and diverges, then $(a_n)$ tends to plus infinity" but "if $(a_n)$ is eventually non-negative and tends to plus infinity, then it does not diverge"?
– UnknownW
Aug 30 at 20:28
@UnknownW: en.wikipedia.org/wiki/Necessity_and_sufficiency
– Yves Daoust
Aug 30 at 21:25
add a comment |Â
What does "necessary" and "sufficient" mean in terms of mathematical symbols? For example, does >"Eventually non-negative" is necessary but not sufficient< mean that "if $(a_n)$ is eventually non-negative and diverges, then $(a_n)$ tends to plus infinity" but "if $(a_n)$ is eventually non-negative and tends to plus infinity, then it does not diverge"?
– UnknownW
Aug 30 at 20:28
@UnknownW: en.wikipedia.org/wiki/Necessity_and_sufficiency
– Yves Daoust
Aug 30 at 21:25
What does "necessary" and "sufficient" mean in terms of mathematical symbols? For example, does >"Eventually non-negative" is necessary but not sufficient< mean that "if $(a_n)$ is eventually non-negative and diverges, then $(a_n)$ tends to plus infinity" but "if $(a_n)$ is eventually non-negative and tends to plus infinity, then it does not diverge"?
– UnknownW
Aug 30 at 20:28
What does "necessary" and "sufficient" mean in terms of mathematical symbols? For example, does >"Eventually non-negative" is necessary but not sufficient< mean that "if $(a_n)$ is eventually non-negative and diverges, then $(a_n)$ tends to plus infinity" but "if $(a_n)$ is eventually non-negative and tends to plus infinity, then it does not diverge"?
– UnknownW
Aug 30 at 20:28
@UnknownW: en.wikipedia.org/wiki/Necessity_and_sufficiency
– Yves Daoust
Aug 30 at 21:25
@UnknownW: en.wikipedia.org/wiki/Necessity_and_sufficiency
– Yves Daoust
Aug 30 at 21:25
add a comment |Â
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3
I don't think there's really a way to restate this better than the definition of tending to infinity.
– Ian
Aug 30 at 9:35
For every $R>0$, you should be able to find an $N geq 1$ so that $a_n geq R$ whenever $n geq N$.
– Sobi
Aug 30 at 9:36
Now that you edited point 2 with PLUS infinity: it is often stated in books that monotone divergent sequences tend either to plus or minus infinty. In this case you would just need $a_n$ to be non decreasing
– Nicolò
Aug 30 at 9:42
$lim inf a_n = infty$, but that is just paraphrasing point 2...
– nicomezi
Aug 30 at 9:42
1
"has to be non-negative eventually" Yes. "or increasing eventually" No. Counterexample: $a_n=n+(-1)^n$.
– Did
Aug 30 at 9:57