Does there exist a sequence $(s_n)$ of partial sums of a series $Sigma a_n$ where the series diverges and the...
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Fairly simple question, but I was unable to thing of a good solution.
It is obvious that the sequence can be bounded as it can just be $s_n=(-1)^n$, but I can not figure out how to prove whether or not it can be increasing. My gut instinct is no, but I don't know how I would prove that. Any help would be appreciated.
real-analysis sequences-and-series
asked Dec 17 '18 at 21:54
Mohammed ShahidMohammed Shahid
1457
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add a comment |
$begingroup$
Fairly simple question, but I was unable to thing of a good solution.
It is obvious that the sequence can be bounded as it can just be $s_n=(-1)^n$, but I can not figure out how to prove whether or not it can be increasing. My gut instinct is no, but I don't know how I would prove that. Any help would be appreciated.
real-analysis sequences-and-series
asked Dec 17 '18 at 21:54
Mohammed ShahidMohammed Shahid
1457
$endgroup$
$begingroup$
Any increasing bounded sequence converges, by the monotone convergence theorem. So the sequence $(s_n)_n$ would converge, meaning the series $sum_n a_n$ would too.
$endgroup$
– Clement C.
Dec 17 '18 at 21:56
add a comment |
$begingroup$
Fairly simple question, but I was unable to thing of a good solution.
It is obvious that the sequence can be bounded as it can just be $s_n=(-1)^n$, but I can not figure out how to prove whether or not it can be increasing. My gut instinct is no, but I don't know how I would prove that. Any help would be appreciated.
real-analysis sequences-and-series
asked Dec 17 '18 at 21:54
Mohammed ShahidMohammed Shahid
1457
$endgroup$
Fairly simple question, but I was unable to thing of a good solution.
It is obvious that the sequence can be bounded as it can just be $s_n=(-1)^n$, but I can not figure out how to prove whether or not it can be increasing. My gut instinct is no, but I don't know how I would prove that. Any help would be appreciated.
real-analysis sequences-and-series
real-analysis sequences-and-series
asked Dec 17 '18 at 21:54
Mohammed ShahidMohammed Shahid
1457
asked Dec 17 '18 at 21:54
Mohammed ShahidMohammed Shahid
1457
asked Dec 17 '18 at 21:54
Mohammed ShahidMohammed Shahid
1457
asked Dec 17 '18 at 21:54
Mohammed ShahidMohammed Shahid
1457
asked Dec 17 '18 at 21:54
Mohammed ShahidMohammed Shahid
1457
1457
$begingroup$
Any increasing bounded sequence converges, by the monotone convergence theorem. So the sequence $(s_n)_n$ would converge, meaning the series $sum_n a_n$ would too.
$endgroup$
– Clement C.
Dec 17 '18 at 21:56
add a comment |
$begingroup$
Any increasing bounded sequence converges, by the monotone convergence theorem. So the sequence $(s_n)_n$ would converge, meaning the series $sum_n a_n$ would too.
$endgroup$
– Clement C.
Dec 17 '18 at 21:56
$begingroup$
Any increasing bounded sequence converges, by the monotone convergence theorem. So the sequence $(s_n)_n$ would converge, meaning the series $sum_n a_n$ would too.
$endgroup$
– Clement C.
Dec 17 '18 at 21:56
$begingroup$
Any increasing bounded sequence converges, by the monotone convergence theorem. So the sequence $(s_n)_n$ would converge, meaning the series $sum_n a_n$ would too.
$endgroup$
– Clement C.
Dec 17 '18 at 21:56
add a comment |
2 Answers
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If the sequence $(s_n)$ of partial sums is increasing and bounded (in particular bounded above) it follows that $s_n$ converges to $sup_{k} s_k<infty$. So the series converges.
answered Dec 17 '18 at 21:57
Foobaz JohnFoobaz John
22.8k41452
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No it is not possible indeed if $s_n$ is bounded and increasing it has finite limit, that is converges, and by definition
$$lim_{nto infty} s_n=sum_{nge n_0} a_n$$
Refer to monotone convergence theorem.
answered Dec 17 '18 at 21:56
gimusigimusi
93k84594
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2 Answers
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2 Answers
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If the sequence $(s_n)$ of partial sums is increasing and bounded (in particular bounded above) it follows that $s_n$ converges to $sup_{k} s_k<infty$. So the series converges.
answered Dec 17 '18 at 21:57
Foobaz JohnFoobaz John
22.8k41452
$endgroup$
add a comment |
$begingroup$
If the sequence $(s_n)$ of partial sums is increasing and bounded (in particular bounded above) it follows that $s_n$ converges to $sup_{k} s_k<infty$. So the series converges.
answered Dec 17 '18 at 21:57
Foobaz JohnFoobaz John
22.8k41452
$endgroup$
add a comment |
$begingroup$
If the sequence $(s_n)$ of partial sums is increasing and bounded (in particular bounded above) it follows that $s_n$ converges to $sup_{k} s_k<infty$. So the series converges.
answered Dec 17 '18 at 21:57
Foobaz JohnFoobaz John
22.8k41452
$endgroup$
If the sequence $(s_n)$ of partial sums is increasing and bounded (in particular bounded above) it follows that $s_n$ converges to $sup_{k} s_k<infty$. So the series converges.
answered Dec 17 '18 at 21:57
Foobaz JohnFoobaz John
22.8k41452
answered Dec 17 '18 at 21:57
Foobaz JohnFoobaz John
22.8k41452
answered Dec 17 '18 at 21:57
Foobaz JohnFoobaz John
22.8k41452
answered Dec 17 '18 at 21:57
Foobaz JohnFoobaz John
22.8k41452
22.8k41452
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No it is not possible indeed if $s_n$ is bounded and increasing it has finite limit, that is converges, and by definition
$$lim_{nto infty} s_n=sum_{nge n_0} a_n$$
Refer to monotone convergence theorem.
answered Dec 17 '18 at 21:56
gimusigimusi
93k84594
$endgroup$
add a comment |
$begingroup$
No it is not possible indeed if $s_n$ is bounded and increasing it has finite limit, that is converges, and by definition
$$lim_{nto infty} s_n=sum_{nge n_0} a_n$$
Refer to monotone convergence theorem.
answered Dec 17 '18 at 21:56
gimusigimusi
93k84594
$endgroup$
add a comment |
$begingroup$
No it is not possible indeed if $s_n$ is bounded and increasing it has finite limit, that is converges, and by definition
$$lim_{nto infty} s_n=sum_{nge n_0} a_n$$
Refer to monotone convergence theorem.
answered Dec 17 '18 at 21:56
gimusigimusi
93k84594
$endgroup$
No it is not possible indeed if $s_n$ is bounded and increasing it has finite limit, that is converges, and by definition
$$lim_{nto infty} s_n=sum_{nge n_0} a_n$$
Refer to monotone convergence theorem.
answered Dec 17 '18 at 21:56
gimusigimusi
93k84594
answered Dec 17 '18 at 21:56
gimusigimusi
93k84594
answered Dec 17 '18 at 21:56
gimusigimusi
93k84594
answered Dec 17 '18 at 21:56
gimusigimusi
93k84594
93k84594
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Any increasing bounded sequence converges, by the monotone convergence theorem. So the sequence $(s_n)_n$ would converge, meaning the series $sum_n a_n$ would too.
$endgroup$
– Clement C.
Dec 17 '18 at 21:56