Subsequential Limits of a Conditionally Convergent Sequence
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Let $sum_{x=1}^infty a_n$ be a conditionally convergent series of real numbers. Let $(s_n)$ be the sequence of $n$-th partial sums. Let $S$ be the set of all subsequential limits of $(s_n)subsetmathbb{R}$.
Is there anything that can be said about $S$? Does it equal $mathbb{R}$ by any consequence of the Riemann Series Theorem?
real-analysis sequences-and-series convergence
asked Jan 21 at 22:50
user162520user162520
672315
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add a comment |
$begingroup$
Let $sum_{x=1}^infty a_n$ be a conditionally convergent series of real numbers. Let $(s_n)$ be the sequence of $n$-th partial sums. Let $S$ be the set of all subsequential limits of $(s_n)subsetmathbb{R}$.
Is there anything that can be said about $S$? Does it equal $mathbb{R}$ by any consequence of the Riemann Series Theorem?
real-analysis sequences-and-series convergence
asked Jan 21 at 22:50
user162520user162520
672315
$endgroup$
$begingroup$
My issue is that a reordering doesn't really count as a subsequence, does it?
$endgroup$
– user162520
Jan 21 at 22:57
add a comment |
$begingroup$
Let $sum_{x=1}^infty a_n$ be a conditionally convergent series of real numbers. Let $(s_n)$ be the sequence of $n$-th partial sums. Let $S$ be the set of all subsequential limits of $(s_n)subsetmathbb{R}$.
Is there anything that can be said about $S$? Does it equal $mathbb{R}$ by any consequence of the Riemann Series Theorem?
real-analysis sequences-and-series convergence
asked Jan 21 at 22:50
user162520user162520
672315
$endgroup$
Let $sum_{x=1}^infty a_n$ be a conditionally convergent series of real numbers. Let $(s_n)$ be the sequence of $n$-th partial sums. Let $S$ be the set of all subsequential limits of $(s_n)subsetmathbb{R}$.
Is there anything that can be said about $S$? Does it equal $mathbb{R}$ by any consequence of the Riemann Series Theorem?
real-analysis sequences-and-series convergence
real-analysis sequences-and-series convergence
asked Jan 21 at 22:50
user162520user162520
672315
asked Jan 21 at 22:50
user162520user162520
672315
asked Jan 21 at 22:50
user162520user162520
672315
asked Jan 21 at 22:50
user162520user162520
672315
asked Jan 21 at 22:50
user162520user162520
672315
672315
$begingroup$
My issue is that a reordering doesn't really count as a subsequence, does it?
$endgroup$
– user162520
Jan 21 at 22:57
add a comment |
$begingroup$
My issue is that a reordering doesn't really count as a subsequence, does it?
$endgroup$
– user162520
Jan 21 at 22:57
$begingroup$
My issue is that a reordering doesn't really count as a subsequence, does it?
$endgroup$
– user162520
Jan 21 at 22:57
$begingroup$
My issue is that a reordering doesn't really count as a subsequence, does it?
$endgroup$
– user162520
Jan 21 at 22:57
add a comment |
1 Answer
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$s_n$ tends to a finite limit $s$. So all subsequence of ${s_n}$ converge to $s$ and the set of limits of subsequences is ${s}$.
answered Jan 21 at 23:49
Kavi Rama MurthyKavi Rama Murthy
62.7k42262
$endgroup$
$begingroup$
Why does $s_n$ tend to a finite limit? Doesn't only $|s_n|$ converge?
$endgroup$
– user162520
Jan 21 at 23:58
$begingroup$
By definition a series converges iff the sequence of partial sums ${s_n}$ tends to a finite limit. In this case the series is given to be convergent. It is not absolutely convergent, but that has no consequence.
$endgroup$
– Kavi Rama Murthy
Jan 22 at 0:00
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$s_n$ tends to a finite limit $s$. So all subsequence of ${s_n}$ converge to $s$ and the set of limits of subsequences is ${s}$.
answered Jan 21 at 23:49
Kavi Rama MurthyKavi Rama Murthy
62.7k42262
$endgroup$
$begingroup$
Why does $s_n$ tend to a finite limit? Doesn't only $|s_n|$ converge?
$endgroup$
– user162520
Jan 21 at 23:58
$begingroup$
By definition a series converges iff the sequence of partial sums ${s_n}$ tends to a finite limit. In this case the series is given to be convergent. It is not absolutely convergent, but that has no consequence.
$endgroup$
– Kavi Rama Murthy
Jan 22 at 0:00
add a comment |
$begingroup$
$s_n$ tends to a finite limit $s$. So all subsequence of ${s_n}$ converge to $s$ and the set of limits of subsequences is ${s}$.
answered Jan 21 at 23:49
Kavi Rama MurthyKavi Rama Murthy
62.7k42262
$endgroup$
$begingroup$
Why does $s_n$ tend to a finite limit? Doesn't only $|s_n|$ converge?
$endgroup$
– user162520
Jan 21 at 23:58
$begingroup$
By definition a series converges iff the sequence of partial sums ${s_n}$ tends to a finite limit. In this case the series is given to be convergent. It is not absolutely convergent, but that has no consequence.
$endgroup$
– Kavi Rama Murthy
Jan 22 at 0:00
add a comment |
$begingroup$
$s_n$ tends to a finite limit $s$. So all subsequence of ${s_n}$ converge to $s$ and the set of limits of subsequences is ${s}$.
answered Jan 21 at 23:49
Kavi Rama MurthyKavi Rama Murthy
62.7k42262
$endgroup$
$s_n$ tends to a finite limit $s$. So all subsequence of ${s_n}$ converge to $s$ and the set of limits of subsequences is ${s}$.
answered Jan 21 at 23:49
Kavi Rama MurthyKavi Rama Murthy
62.7k42262
answered Jan 21 at 23:49
Kavi Rama MurthyKavi Rama Murthy
62.7k42262
answered Jan 21 at 23:49
Kavi Rama MurthyKavi Rama Murthy
62.7k42262
answered Jan 21 at 23:49
Kavi Rama MurthyKavi Rama Murthy
62.7k42262
62.7k42262
$begingroup$
Why does $s_n$ tend to a finite limit? Doesn't only $|s_n|$ converge?
$endgroup$
– user162520
Jan 21 at 23:58
$begingroup$
By definition a series converges iff the sequence of partial sums ${s_n}$ tends to a finite limit. In this case the series is given to be convergent. It is not absolutely convergent, but that has no consequence.
$endgroup$
– Kavi Rama Murthy
Jan 22 at 0:00
add a comment |
$begingroup$
Why does $s_n$ tend to a finite limit? Doesn't only $|s_n|$ converge?
$endgroup$
– user162520
Jan 21 at 23:58
$begingroup$
By definition a series converges iff the sequence of partial sums ${s_n}$ tends to a finite limit. In this case the series is given to be convergent. It is not absolutely convergent, but that has no consequence.
$endgroup$
– Kavi Rama Murthy
Jan 22 at 0:00
$begingroup$
Why does $s_n$ tend to a finite limit? Doesn't only $|s_n|$ converge?
$endgroup$
– user162520
Jan 21 at 23:58
$begingroup$
Why does $s_n$ tend to a finite limit? Doesn't only $|s_n|$ converge?
$endgroup$
– user162520
Jan 21 at 23:58
$begingroup$
By definition a series converges iff the sequence of partial sums ${s_n}$ tends to a finite limit. In this case the series is given to be convergent. It is not absolutely convergent, but that has no consequence.
$endgroup$
– Kavi Rama Murthy
Jan 22 at 0:00
$begingroup$
By definition a series converges iff the sequence of partial sums ${s_n}$ tends to a finite limit. In this case the series is given to be convergent. It is not absolutely convergent, but that has no consequence.
$endgroup$
– Kavi Rama Murthy
Jan 22 at 0:00
add a comment |
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$begingroup$
My issue is that a reordering doesn't really count as a subsequence, does it?
$endgroup$
– user162520
Jan 21 at 22:57