'Sum' of another infinite series. [closed]
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So my wording may not he exactly right upon this matter.
(Note; wherever i use {...} it means the series goes on infinitely many times.)
I will be attempting to show the value of sum of
n+n+n+...
To do so; i assume the 'sum' of the grandi series(S')
1-1+1-1+... =1/2
And i denote
S=1+1+1+...
S+S'=2+2+2+2+... (the odd terms add up and the even terms cancel out;I believe the {...} part shouldn't be affected)
S+S'=2S
Hence, S=S'=1/2
Therefore;
n+n+n+...=n(1+1+1+...)=n(1/2)
=>n+n+n+...=n/2
Now i know, that S and S' are diverging series but there is something called the Cesàro summation? The main doubt i have; is regarding the fact (n+n+n+...)=n/2 i feel its somewhere wrong however i cannot seem to find where it might bw. Anyway i would also like to gain insight on the matter and would like to know more about these kind of 'summations'
number-theory
asked Nov 16 at 23:49
user182947
164
closed as unclear what you're asking by Servaes, Kavi Rama Murthy, Chinnapparaj R, Brahadeesh, Lord Shark the Unknown Nov 17 at 6:00
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
up vote
-1
down vote
favorite
So my wording may not he exactly right upon this matter.
(Note; wherever i use {...} it means the series goes on infinitely many times.)
I will be attempting to show the value of sum of
n+n+n+...
To do so; i assume the 'sum' of the grandi series(S')
1-1+1-1+... =1/2
And i denote
S=1+1+1+...
S+S'=2+2+2+2+... (the odd terms add up and the even terms cancel out;I believe the {...} part shouldn't be affected)
S+S'=2S
Hence, S=S'=1/2
Therefore;
n+n+n+...=n(1+1+1+...)=n(1/2)
=>n+n+n+...=n/2
Now i know, that S and S' are diverging series but there is something called the Cesàro summation? The main doubt i have; is regarding the fact (n+n+n+...)=n/2 i feel its somewhere wrong however i cannot seem to find where it might bw. Anyway i would also like to gain insight on the matter and would like to know more about these kind of 'summations'
number-theory
asked Nov 16 at 23:49
user182947
164
closed as unclear what you're asking by Servaes, Kavi Rama Murthy, Chinnapparaj R, Brahadeesh, Lord Shark the Unknown Nov 17 at 6:00
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
$texttt{MathJax}$ Basic Tutorial and Quick Reference
– Felix Marin
Nov 17 at 0:09
This Mathologer video can be a good beginning.
– rafa11111
Nov 17 at 0:31
Up to some $infty$ constant, $1 + 1 + 1 + cdots = -frac12 implies 2 + 0 + 2 + 0 + cdots = 0 ne -1 = 2 + 2 + 2 + cdots$.
– achille hui
Nov 17 at 0:34
Is my approach correct?
– user182947
Nov 17 at 1:41
You can't manipulate infinite series like that.
– Henrik
Nov 17 at 16:26
add a comment |
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
So my wording may not he exactly right upon this matter.
(Note; wherever i use {...} it means the series goes on infinitely many times.)
I will be attempting to show the value of sum of
n+n+n+...
To do so; i assume the 'sum' of the grandi series(S')
1-1+1-1+... =1/2
And i denote
S=1+1+1+...
S+S'=2+2+2+2+... (the odd terms add up and the even terms cancel out;I believe the {...} part shouldn't be affected)
S+S'=2S
Hence, S=S'=1/2
Therefore;
n+n+n+...=n(1+1+1+...)=n(1/2)
=>n+n+n+...=n/2
Now i know, that S and S' are diverging series but there is something called the Cesàro summation? The main doubt i have; is regarding the fact (n+n+n+...)=n/2 i feel its somewhere wrong however i cannot seem to find where it might bw. Anyway i would also like to gain insight on the matter and would like to know more about these kind of 'summations'
number-theory
asked Nov 16 at 23:49
user182947
164
So my wording may not he exactly right upon this matter.
(Note; wherever i use {...} it means the series goes on infinitely many times.)
I will be attempting to show the value of sum of
n+n+n+...
To do so; i assume the 'sum' of the grandi series(S')
1-1+1-1+... =1/2
And i denote
S=1+1+1+...
S+S'=2+2+2+2+... (the odd terms add up and the even terms cancel out;I believe the {...} part shouldn't be affected)
S+S'=2S
Hence, S=S'=1/2
Therefore;
n+n+n+...=n(1+1+1+...)=n(1/2)
=>n+n+n+...=n/2
Now i know, that S and S' are diverging series but there is something called the Cesàro summation? The main doubt i have; is regarding the fact (n+n+n+...)=n/2 i feel its somewhere wrong however i cannot seem to find where it might bw. Anyway i would also like to gain insight on the matter and would like to know more about these kind of 'summations'
number-theory
number-theory
asked Nov 16 at 23:49
user182947
164
asked Nov 16 at 23:49
user182947
164
edited Nov 17 at 15:38
asked Nov 16 at 23:49
user182947
164
asked Nov 16 at 23:49
user182947
164
asked Nov 16 at 23:49
user182947
164
164
closed as unclear what you're asking by Servaes, Kavi Rama Murthy, Chinnapparaj R, Brahadeesh, Lord Shark the Unknown Nov 17 at 6:00
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as unclear what you're asking by Servaes, Kavi Rama Murthy, Chinnapparaj R, Brahadeesh, Lord Shark the Unknown Nov 17 at 6:00
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
$texttt{MathJax}$ Basic Tutorial and Quick Reference
– Felix Marin
Nov 17 at 0:09
This Mathologer video can be a good beginning.
– rafa11111
Nov 17 at 0:31
Up to some $infty$ constant, $1 + 1 + 1 + cdots = -frac12 implies 2 + 0 + 2 + 0 + cdots = 0 ne -1 = 2 + 2 + 2 + cdots$.
– achille hui
Nov 17 at 0:34
Is my approach correct?
– user182947
Nov 17 at 1:41
You can't manipulate infinite series like that.
– Henrik
Nov 17 at 16:26
add a comment |
$texttt{MathJax}$ Basic Tutorial and Quick Reference
– Felix Marin
Nov 17 at 0:09
This Mathologer video can be a good beginning.
– rafa11111
Nov 17 at 0:31
Up to some $infty$ constant, $1 + 1 + 1 + cdots = -frac12 implies 2 + 0 + 2 + 0 + cdots = 0 ne -1 = 2 + 2 + 2 + cdots$.
– achille hui
Nov 17 at 0:34
Is my approach correct?
– user182947
Nov 17 at 1:41
You can't manipulate infinite series like that.
– Henrik
Nov 17 at 16:26
$texttt{MathJax}$ Basic Tutorial and Quick Reference
– Felix Marin
Nov 17 at 0:09
$texttt{MathJax}$ Basic Tutorial and Quick Reference
– Felix Marin
Nov 17 at 0:09
This Mathologer video can be a good beginning.
– rafa11111
Nov 17 at 0:31
This Mathologer video can be a good beginning.
– rafa11111
Nov 17 at 0:31
Up to some $infty$ constant, $1 + 1 + 1 + cdots = -frac12 implies 2 + 0 + 2 + 0 + cdots = 0 ne -1 = 2 + 2 + 2 + cdots$.
– achille hui
Nov 17 at 0:34
Up to some $infty$ constant, $1 + 1 + 1 + cdots = -frac12 implies 2 + 0 + 2 + 0 + cdots = 0 ne -1 = 2 + 2 + 2 + cdots$.
– achille hui
Nov 17 at 0:34
Is my approach correct?
– user182947
Nov 17 at 1:41
Is my approach correct?
– user182947
Nov 17 at 1:41
You can't manipulate infinite series like that.
– Henrik
Nov 17 at 16:26
You can't manipulate infinite series like that.
– Henrik
Nov 17 at 16:26
add a comment |
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$texttt{MathJax}$ Basic Tutorial and Quick Reference
– Felix Marin
Nov 17 at 0:09
This Mathologer video can be a good beginning.
– rafa11111
Nov 17 at 0:31
Up to some $infty$ constant, $1 + 1 + 1 + cdots = -frac12 implies 2 + 0 + 2 + 0 + cdots = 0 ne -1 = 2 + 2 + 2 + cdots$.
– achille hui
Nov 17 at 0:34
Is my approach correct?
– user182947
Nov 17 at 1:41
You can't manipulate infinite series like that.
– Henrik
Nov 17 at 16:26