How do we define summation over an arbitrary index set including negative values?
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
1,080615
|
show 1 more comment
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
1,080615
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
1,080615
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
1,080615
asked Nov 21 at 4:27
伽罗瓦
1,080615
asked Nov 21 at 4:27
伽罗瓦
1,080615
asked Nov 21 at 4:27
伽罗瓦
1,080615
asked Nov 21 at 4:27
伽罗瓦
1,080615
1,080615
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
1
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
7787
add a comment |
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
7787
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
7787
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
7787
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
7787
answered Nov 21 at 5:09
Dante Grevino
7787
answered Nov 21 at 5:09
Dante Grevino
7787
answered Nov 21 at 5:09
Dante Grevino
7787
7787
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3007259%2fhow-do-we-define-summation-over-an-arbitrary-index-set-including-negative-values%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
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