Countability of a set of subsequences .
2
$begingroup$
Consider a sequence $x_n$ with positive values such that
$sum _{n=1}^{infty} x_n $ converges .
Is the set of the subsequences $ x_{k_n}$ of $x_n$ such that
$sum _{n=1}^{infty} x_{k_n} =c in R$ countable ?
$c$ is previously fixed so the corresponding series of every subsequence converges to the same value.
analysis
asked Dec 11 '18 at 17:39
mike mokemike moke
406
$endgroup$
add a comment |
2
$begingroup$
Consider a sequence $x_n$ with positive values such that
$sum _{n=1}^{infty} x_n $ converges .
Is the set of the subsequences $ x_{k_n}$ of $x_n$ such that
$sum _{n=1}^{infty} x_{k_n} =c in R$ countable ?
$c$ is previously fixed so the corresponding series of every subsequence converges to the same value.
analysis
asked Dec 11 '18 at 17:39
mike mokemike moke
406
$endgroup$
$begingroup$
I have answered assuming that you mean the set of subsequences such that the corresponding series converges. Do you mean that $c$ is previoulsy fixed instead? That is, your question is that, given $cin Bbb R$, the set of subsequences such that $sum x_{k_n}=c$ is countable?
$endgroup$
– ajotatxe
Dec 11 '18 at 17:53
$begingroup$
Yes I mean that c if previously fixed .So for all the subsequences their corresponding series converges to the same c in $R$
$endgroup$
– mike moke
Dec 11 '18 at 17:57
$begingroup$
Then the answer depends clearly on $c$. For example, if $c>sum x_n$, the set is void. If $c=sum x_n$ the set is a singleton. For intermediate values, it depends also on the terms of the sequence.
$endgroup$
– ajotatxe
Dec 11 '18 at 19:13
$begingroup$
In both cases the set is countable .If the answer is no then we need to construct an uncountable set .
$endgroup$
– mike moke
Dec 11 '18 at 19:17
add a comment |
2
2
2
1
$begingroup$
Consider a sequence $x_n$ with positive values such that
$sum _{n=1}^{infty} x_n $ converges .
Is the set of the subsequences $ x_{k_n}$ of $x_n$ such that
$sum _{n=1}^{infty} x_{k_n} =c in R$ countable ?
$c$ is previously fixed so the corresponding series of every subsequence converges to the same value.
analysis
asked Dec 11 '18 at 17:39
mike mokemike moke
406
$endgroup$
Consider a sequence $x_n$ with positive values such that
$sum _{n=1}^{infty} x_n $ converges .
Is the set of the subsequences $ x_{k_n}$ of $x_n$ such that
$sum _{n=1}^{infty} x_{k_n} =c in R$ countable ?
$c$ is previously fixed so the corresponding series of every subsequence converges to the same value.
analysis
analysis
asked Dec 11 '18 at 17:39
mike mokemike moke
406
asked Dec 11 '18 at 17:39
mike mokemike moke
406
edited Dec 11 '18 at 18:13
mike moke
asked Dec 11 '18 at 17:39
mike mokemike moke
406
asked Dec 11 '18 at 17:39
mike mokemike moke
406
asked Dec 11 '18 at 17:39
mike mokemike moke
406
406
$begingroup$
I have answered assuming that you mean the set of subsequences such that the corresponding series converges. Do you mean that $c$ is previoulsy fixed instead? That is, your question is that, given $cin Bbb R$, the set of subsequences such that $sum x_{k_n}=c$ is countable?
$endgroup$
– ajotatxe
Dec 11 '18 at 17:53
$begingroup$
Yes I mean that c if previously fixed .So for all the subsequences their corresponding series converges to the same c in $R$
$endgroup$
– mike moke
Dec 11 '18 at 17:57
$begingroup$
Then the answer depends clearly on $c$. For example, if $c>sum x_n$, the set is void. If $c=sum x_n$ the set is a singleton. For intermediate values, it depends also on the terms of the sequence.
$endgroup$
– ajotatxe
Dec 11 '18 at 19:13
$begingroup$
In both cases the set is countable .If the answer is no then we need to construct an uncountable set .
$endgroup$
– mike moke
Dec 11 '18 at 19:17
add a comment |
$begingroup$
I have answered assuming that you mean the set of subsequences such that the corresponding series converges. Do you mean that $c$ is previoulsy fixed instead? That is, your question is that, given $cin Bbb R$, the set of subsequences such that $sum x_{k_n}=c$ is countable?
$endgroup$
– ajotatxe
Dec 11 '18 at 17:53
$begingroup$
Yes I mean that c if previously fixed .So for all the subsequences their corresponding series converges to the same c in $R$
$endgroup$
– mike moke
Dec 11 '18 at 17:57
$begingroup$
Then the answer depends clearly on $c$. For example, if $c>sum x_n$, the set is void. If $c=sum x_n$ the set is a singleton. For intermediate values, it depends also on the terms of the sequence.
$endgroup$
– ajotatxe
Dec 11 '18 at 19:13
$begingroup$
In both cases the set is countable .If the answer is no then we need to construct an uncountable set .
$endgroup$
– mike moke
Dec 11 '18 at 19:17
$begingroup$
I have answered assuming that you mean the set of subsequences such that the corresponding series converges. Do you mean that $c$ is previoulsy fixed instead? That is, your question is that, given $cin Bbb R$, the set of subsequences such that $sum x_{k_n}=c$ is countable?
$endgroup$
– ajotatxe
Dec 11 '18 at 17:53
$begingroup$
I have answered assuming that you mean the set of subsequences such that the corresponding series converges. Do you mean that $c$ is previoulsy fixed instead? That is, your question is that, given $cin Bbb R$, the set of subsequences such that $sum x_{k_n}=c$ is countable?
$endgroup$
– ajotatxe
Dec 11 '18 at 17:53
$begingroup$
Yes I mean that c if previously fixed .So for all the subsequences their corresponding series converges to the same c in $R$
$endgroup$
– mike moke
Dec 11 '18 at 17:57
$begingroup$
Yes I mean that c if previously fixed .So for all the subsequences their corresponding series converges to the same c in $R$
$endgroup$
– mike moke
Dec 11 '18 at 17:57
$begingroup$
Then the answer depends clearly on $c$. For example, if $c>sum x_n$, the set is void. If $c=sum x_n$ the set is a singleton. For intermediate values, it depends also on the terms of the sequence.
$endgroup$
– ajotatxe
Dec 11 '18 at 19:13
$begingroup$
Then the answer depends clearly on $c$. For example, if $c>sum x_n$, the set is void. If $c=sum x_n$ the set is a singleton. For intermediate values, it depends also on the terms of the sequence.
$endgroup$
– ajotatxe
Dec 11 '18 at 19:13
$begingroup$
In both cases the set is countable .If the answer is no then we need to construct an uncountable set .
$endgroup$
– mike moke
Dec 11 '18 at 19:17
$begingroup$
In both cases the set is countable .If the answer is no then we need to construct an uncountable set .
$endgroup$
– mike moke
Dec 11 '18 at 19:17
add a comment |
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$begingroup$
I have answered assuming that you mean the set of subsequences such that the corresponding series converges. Do you mean that $c$ is previoulsy fixed instead? That is, your question is that, given $cin Bbb R$, the set of subsequences such that $sum x_{k_n}=c$ is countable?
$endgroup$
– ajotatxe
Dec 11 '18 at 17:53
$begingroup$
Yes I mean that c if previously fixed .So for all the subsequences their corresponding series converges to the same c in $R$
$endgroup$
– mike moke
Dec 11 '18 at 17:57
$begingroup$
Then the answer depends clearly on $c$. For example, if $c>sum x_n$, the set is void. If $c=sum x_n$ the set is a singleton. For intermediate values, it depends also on the terms of the sequence.
$endgroup$
– ajotatxe
Dec 11 '18 at 19:13
$begingroup$
In both cases the set is countable .If the answer is no then we need to construct an uncountable set .
$endgroup$
– mike moke
Dec 11 '18 at 19:17