How do we define summation over an arbitrary index set including negative values?
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I just read that for $f: I rightarrow [0, infty),$ we can define $sum_{i in I} f(i)$ as $sup{sum_{i in F} f(i) mid F subset I, F$ finite $}.$ It makes sense that we can define it in this way. It coincides with my notion for countable sums. However, how can we define such a sum where $f$ may include negative values. It suddenly does not make sense to take the supremum.
That is, if $I$ is an arbitrary set and $x_i in mathbb{R},$ how would we define $sum_{i in I} x_i?$
summation
asked Nov 21 at 4:27
伽罗瓦
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I just read that for $f: I rightarrow [0, infty),$ we can define $sum_{i in I} f(i)$ as $sup{sum_{i in F} f(i) mid F subset I, F$ finite $}.$ It makes sense that we can define it in this way. It coincides with my notion for countable sums. However, how can we define such a sum where $f$ may include negative values. It suddenly does not make sense to take the supremum.
That is, if $I$ is an arbitrary set and $x_i in mathbb{R},$ how would we define $sum_{i in I} x_i?$
summation
asked Nov 21 at 4:27
伽罗瓦
1,068615
Still the same way as above since the arbitraryness (if I may use the word) is in the index an not the values having the index.
– Yadati Kiran
Nov 21 at 4:31
3
Short answer: in general, you can't. For instance, conditionally convergent sums can be rearranged to sum to any given number, so if $a_n$ converges conditionally and $I = mathbb{N}$, then $sum_{i in mathbb{N}} a_i$ isn't uniquely defined, because it's order-dependent. An interesting question to ask is, roughly speaking, how "many" negative terms can be included in the sequence before things begin to go awry.
– MathematicsStudent1122
Nov 21 at 4:31
@YadatiKiran but we are taking the supremum of the sum itself - so if $f$ is purely a negative valued function then taking the supremum of the finite sums would not work I think.
– 伽罗瓦
Nov 21 at 4:32
@伽罗瓦: As MathematicsStudent1122 suggests we see" how "many" negative terms can be included in the sequence before things begin to go awry."
– Yadati Kiran
Nov 21 at 4:36
Just think about the countable case. Supremum of finite sums if of no use and trying to assign some value to the sum of an arbitrary sequence of real numbers is an exercise in futility.
– Kavi Rama Murthy
Nov 21 at 4:53
|
show 1 more comment
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I just read that for $f: I rightarrow [0, infty),$ we can define $sum_{i in I} f(i)$ as $sup{sum_{i in F} f(i) mid F subset I, F$ finite $}.$ It makes sense that we can define it in this way. It coincides with my notion for countable sums. However, how can we define such a sum where $f$ may include negative values. It suddenly does not make sense to take the supremum.
That is, if $I$ is an arbitrary set and $x_i in mathbb{R},$ how would we define $sum_{i in I} x_i?$
summation
asked Nov 21 at 4:27
伽罗瓦
1,068615
I just read that for $f: I rightarrow [0, infty),$ we can define $sum_{i in I} f(i)$ as $sup{sum_{i in F} f(i) mid F subset I, F$ finite $}.$ It makes sense that we can define it in this way. It coincides with my notion for countable sums. However, how can we define such a sum where $f$ may include negative values. It suddenly does not make sense to take the supremum.
That is, if $I$ is an arbitrary set and $x_i in mathbb{R},$ how would we define $sum_{i in I} x_i?$
summation
summation
asked Nov 21 at 4:27
伽罗瓦
1,068615
asked Nov 21 at 4:27
伽罗瓦
1,068615
asked Nov 21 at 4:27
伽罗瓦
1,068615
asked Nov 21 at 4:27
伽罗瓦
1,068615
asked Nov 21 at 4:27
伽罗瓦
1,068615
1,068615
Still the same way as above since the arbitraryness (if I may use the word) is in the index an not the values having the index.
– Yadati Kiran
Nov 21 at 4:31
3
Short answer: in general, you can't. For instance, conditionally convergent sums can be rearranged to sum to any given number, so if $a_n$ converges conditionally and $I = mathbb{N}$, then $sum_{i in mathbb{N}} a_i$ isn't uniquely defined, because it's order-dependent. An interesting question to ask is, roughly speaking, how "many" negative terms can be included in the sequence before things begin to go awry.
– MathematicsStudent1122
Nov 21 at 4:31
@YadatiKiran but we are taking the supremum of the sum itself - so if $f$ is purely a negative valued function then taking the supremum of the finite sums would not work I think.
– 伽罗瓦
Nov 21 at 4:32
@伽罗瓦: As MathematicsStudent1122 suggests we see" how "many" negative terms can be included in the sequence before things begin to go awry."
– Yadati Kiran
Nov 21 at 4:36
Just think about the countable case. Supremum of finite sums if of no use and trying to assign some value to the sum of an arbitrary sequence of real numbers is an exercise in futility.
– Kavi Rama Murthy
Nov 21 at 4:53
|
show 1 more comment
Still the same way as above since the arbitraryness (if I may use the word) is in the index an not the values having the index.
– Yadati Kiran
Nov 21 at 4:31
3
Short answer: in general, you can't. For instance, conditionally convergent sums can be rearranged to sum to any given number, so if $a_n$ converges conditionally and $I = mathbb{N}$, then $sum_{i in mathbb{N}} a_i$ isn't uniquely defined, because it's order-dependent. An interesting question to ask is, roughly speaking, how "many" negative terms can be included in the sequence before things begin to go awry.
– MathematicsStudent1122
Nov 21 at 4:31
@YadatiKiran but we are taking the supremum of the sum itself - so if $f$ is purely a negative valued function then taking the supremum of the finite sums would not work I think.
– 伽罗瓦
Nov 21 at 4:32
@伽罗瓦: As MathematicsStudent1122 suggests we see" how "many" negative terms can be included in the sequence before things begin to go awry."
– Yadati Kiran
Nov 21 at 4:36
Just think about the countable case. Supremum of finite sums if of no use and trying to assign some value to the sum of an arbitrary sequence of real numbers is an exercise in futility.
– Kavi Rama Murthy
Nov 21 at 4:53
Still the same way as above since the arbitraryness (if I may use the word) is in the index an not the values having the index.
– Yadati Kiran
Nov 21 at 4:31
Still the same way as above since the arbitraryness (if I may use the word) is in the index an not the values having the index.
– Yadati Kiran
Nov 21 at 4:31
3
3
Short answer: in general, you can't. For instance, conditionally convergent sums can be rearranged to sum to any given number, so if $a_n$ converges conditionally and $I = mathbb{N}$, then $sum_{i in mathbb{N}} a_i$ isn't uniquely defined, because it's order-dependent. An interesting question to ask is, roughly speaking, how "many" negative terms can be included in the sequence before things begin to go awry.
– MathematicsStudent1122
Nov 21 at 4:31
Short answer: in general, you can't. For instance, conditionally convergent sums can be rearranged to sum to any given number, so if $a_n$ converges conditionally and $I = mathbb{N}$, then $sum_{i in mathbb{N}} a_i$ isn't uniquely defined, because it's order-dependent. An interesting question to ask is, roughly speaking, how "many" negative terms can be included in the sequence before things begin to go awry.
– MathematicsStudent1122
Nov 21 at 4:31
@YadatiKiran but we are taking the supremum of the sum itself - so if $f$ is purely a negative valued function then taking the supremum of the finite sums would not work I think.
– 伽罗瓦
Nov 21 at 4:32
@YadatiKiran but we are taking the supremum of the sum itself - so if $f$ is purely a negative valued function then taking the supremum of the finite sums would not work I think.
– 伽罗瓦
Nov 21 at 4:32
@伽罗瓦: As MathematicsStudent1122 suggests we see" how "many" negative terms can be included in the sequence before things begin to go awry."
– Yadati Kiran
Nov 21 at 4:36
@伽罗瓦: As MathematicsStudent1122 suggests we see" how "many" negative terms can be included in the sequence before things begin to go awry."
– Yadati Kiran
Nov 21 at 4:36
Just think about the countable case. Supremum of finite sums if of no use and trying to assign some value to the sum of an arbitrary sequence of real numbers is an exercise in futility.
– Kavi Rama Murthy
Nov 21 at 4:53
Just think about the countable case. Supremum of finite sums if of no use and trying to assign some value to the sum of an arbitrary sequence of real numbers is an exercise in futility.
– Kavi Rama Murthy
Nov 21 at 4:53
|
show 1 more comment
1 Answer
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Compute the sum over the non-negative values and compute the sum over the non-positive values. If one of them is finite take the sum of the two results. As in the Lebesgue's integral.
answered Nov 21 at 5:09
Dante Grevino
4616
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Compute the sum over the non-negative values and compute the sum over the non-positive values. If one of them is finite take the sum of the two results. As in the Lebesgue's integral.
answered Nov 21 at 5:09
Dante Grevino
4616
New contributor
Dante Grevino is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
up vote
1
down vote
Compute the sum over the non-negative values and compute the sum over the non-positive values. If one of them is finite take the sum of the two results. As in the Lebesgue's integral.
answered Nov 21 at 5:09
Dante Grevino
4616
New contributor
Dante Grevino is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
up vote
1
down vote
up vote
1
down vote
Compute the sum over the non-negative values and compute the sum over the non-positive values. If one of them is finite take the sum of the two results. As in the Lebesgue's integral.
answered Nov 21 at 5:09
Dante Grevino
4616
New contributor
Dante Grevino is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Compute the sum over the non-negative values and compute the sum over the non-positive values. If one of them is finite take the sum of the two results. As in the Lebesgue's integral.
answered Nov 21 at 5:09
Dante Grevino
4616
New contributor
Dante Grevino is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered Nov 21 at 5:09
Dante Grevino
4616
New contributor
Dante Grevino is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered Nov 21 at 5:09
Dante Grevino
4616
answered Nov 21 at 5:09
Dante Grevino
4616
4616
New contributor
Dante Grevino is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Dante Grevino is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Dante Grevino is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
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Still the same way as above since the arbitraryness (if I may use the word) is in the index an not the values having the index.
– Yadati Kiran
Nov 21 at 4:31
3
Short answer: in general, you can't. For instance, conditionally convergent sums can be rearranged to sum to any given number, so if $a_n$ converges conditionally and $I = mathbb{N}$, then $sum_{i in mathbb{N}} a_i$ isn't uniquely defined, because it's order-dependent. An interesting question to ask is, roughly speaking, how "many" negative terms can be included in the sequence before things begin to go awry.
– MathematicsStudent1122
Nov 21 at 4:31
@YadatiKiran but we are taking the supremum of the sum itself - so if $f$ is purely a negative valued function then taking the supremum of the finite sums would not work I think.
– 伽罗瓦
Nov 21 at 4:32
@伽罗瓦: As MathematicsStudent1122 suggests we see" how "many" negative terms can be included in the sequence before things begin to go awry."
– Yadati Kiran
Nov 21 at 4:36
Just think about the countable case. Supremum of finite sums if of no use and trying to assign some value to the sum of an arbitrary sequence of real numbers is an exercise in futility.
– Kavi Rama Murthy
Nov 21 at 4:53