Summation with index in the middle of the summand. What does it mean?

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











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enter image description here



I don't know what the summation in the above pic is saying.



Assume that the x's (1 to n) shown above are numbers that have already been assigned values. n is a fixed value.



(I found this in Atiyah-Macdonald in a proof of some proposition. I tried looking at alternate proofs of the proposition online seeking an explanation for what the summation notation meant but all the proofs were really different from the one in the book.)




commutative-algebra notation




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  • 1




    I don't know exactly what you're asking but note that $x_i$ is missing which is one of possible reasons to emphasize "the middle" like this.
    – zzuussee
    Aug 11 at 20:40










  • This means: $(x_2x_3ldots x_n)+(x_1x_3ldots x_n)+(x_1x_2x_4ldots x_n)+dotsb + (x_1x_2ldots x_n-1)$.
    – Anurag A
    Aug 11 at 20:43














up vote
2
down vote

favorite












enter image description here



I don't know what the summation in the above pic is saying.



Assume that the x's (1 to n) shown above are numbers that have already been assigned values. n is a fixed value.



(I found this in Atiyah-Macdonald in a proof of some proposition. I tried looking at alternate proofs of the proposition online seeking an explanation for what the summation notation meant but all the proofs were really different from the one in the book.)




commutative-algebra notation




share|cite|improve this question


















  • 1




    I don't know exactly what you're asking but note that $x_i$ is missing which is one of possible reasons to emphasize "the middle" like this.
    – zzuussee
    Aug 11 at 20:40










  • This means: $(x_2x_3ldots x_n)+(x_1x_3ldots x_n)+(x_1x_2x_4ldots x_n)+dotsb + (x_1x_2ldots x_n-1)$.
    – Anurag A
    Aug 11 at 20:43












up vote
2
down vote

favorite









up vote
2
down vote

favorite











enter image description here



I don't know what the summation in the above pic is saying.



Assume that the x's (1 to n) shown above are numbers that have already been assigned values. n is a fixed value.



(I found this in Atiyah-Macdonald in a proof of some proposition. I tried looking at alternate proofs of the proposition online seeking an explanation for what the summation notation meant but all the proofs were really different from the one in the book.)




commutative-algebra notation




share|cite|improve this question














enter image description here



I don't know what the summation in the above pic is saying.



Assume that the x's (1 to n) shown above are numbers that have already been assigned values. n is a fixed value.



(I found this in Atiyah-Macdonald in a proof of some proposition. I tried looking at alternate proofs of the proposition online seeking an explanation for what the summation notation meant but all the proofs were really different from the one in the book.)





commutative-algebra notation





share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 11 at 20:48

























asked Aug 11 at 20:39









Barycentric_Bash

31128




31128







  • 1




    I don't know exactly what you're asking but note that $x_i$ is missing which is one of possible reasons to emphasize "the middle" like this.
    – zzuussee
    Aug 11 at 20:40










  • This means: $(x_2x_3ldots x_n)+(x_1x_3ldots x_n)+(x_1x_2x_4ldots x_n)+dotsb + (x_1x_2ldots x_n-1)$.
    – Anurag A
    Aug 11 at 20:43












  • 1




    I don't know exactly what you're asking but note that $x_i$ is missing which is one of possible reasons to emphasize "the middle" like this.
    – zzuussee
    Aug 11 at 20:40










  • This means: $(x_2x_3ldots x_n)+(x_1x_3ldots x_n)+(x_1x_2x_4ldots x_n)+dotsb + (x_1x_2ldots x_n-1)$.
    – Anurag A
    Aug 11 at 20:43







1




1




I don't know exactly what you're asking but note that $x_i$ is missing which is one of possible reasons to emphasize "the middle" like this.
– zzuussee
Aug 11 at 20:40




I don't know exactly what you're asking but note that $x_i$ is missing which is one of possible reasons to emphasize "the middle" like this.
– zzuussee
Aug 11 at 20:40












This means: $(x_2x_3ldots x_n)+(x_1x_3ldots x_n)+(x_1x_2x_4ldots x_n)+dotsb + (x_1x_2ldots x_n-1)$.
– Anurag A
Aug 11 at 20:43




This means: $(x_2x_3ldots x_n)+(x_1x_3ldots x_n)+(x_1x_2x_4ldots x_n)+dotsb + (x_1x_2ldots x_n-1)$.
– Anurag A
Aug 11 at 20:43










1 Answer
1






active

oldest

votes

















up vote
3
down vote



accepted










Note, that at every summation step $i$, $x_i$ is missing in the product, so I'd suggest that, as the index counts upwards, the missed out value in the product moves through the $x_i's$, i.e.



$$sum_i=1^nx_1x_2dots x_i-1x_i+1dots x_n= (x_2dots x_n) + (x_1x_3dots x_n)+dots +(x_1dots x_n-1)$$



Or, in a little more compact notation:



$$sum_i=1^nx_1x_2dots x_i-1x_i+1dots x_n=sum_i=1^nprod_jleq n, jneq ix_j=prod_jleq n, jneq 1x_j+dots +prod_jleq n, jneq nx_j$$






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  • I know it isn't proper etiquette to respond with non-constructive comments, but cheers! I also need to rethink my life choices...
    – Barycentric_Bash
    Aug 11 at 20:50










  • @Barycentric_Bash To continue "non-proper etiquette", let me ask, why do you need to rethink you life choices?
    – zzuussee
    Aug 11 at 20:51











  • I have to confess that I spent upwards of ~30 min trying to find a 'solution' to my problem because i didn't see the missing xi
    – Barycentric_Bash
    Aug 11 at 20:59






  • 3




    @Barycentric_Bash Notation is the burden all of us have to carry when doing mathematics. The foremost important thing is to understand. So, I think, don't worry to much about that.
    – zzuussee
    Aug 11 at 21:01










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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
3
down vote



accepted










Note, that at every summation step $i$, $x_i$ is missing in the product, so I'd suggest that, as the index counts upwards, the missed out value in the product moves through the $x_i's$, i.e.



$$sum_i=1^nx_1x_2dots x_i-1x_i+1dots x_n= (x_2dots x_n) + (x_1x_3dots x_n)+dots +(x_1dots x_n-1)$$



Or, in a little more compact notation:



$$sum_i=1^nx_1x_2dots x_i-1x_i+1dots x_n=sum_i=1^nprod_jleq n, jneq ix_j=prod_jleq n, jneq 1x_j+dots +prod_jleq n, jneq nx_j$$






share|cite|improve this answer






















  • I know it isn't proper etiquette to respond with non-constructive comments, but cheers! I also need to rethink my life choices...
    – Barycentric_Bash
    Aug 11 at 20:50










  • @Barycentric_Bash To continue "non-proper etiquette", let me ask, why do you need to rethink you life choices?
    – zzuussee
    Aug 11 at 20:51











  • I have to confess that I spent upwards of ~30 min trying to find a 'solution' to my problem because i didn't see the missing xi
    – Barycentric_Bash
    Aug 11 at 20:59






  • 3




    @Barycentric_Bash Notation is the burden all of us have to carry when doing mathematics. The foremost important thing is to understand. So, I think, don't worry to much about that.
    – zzuussee
    Aug 11 at 21:01














up vote
3
down vote



accepted










Note, that at every summation step $i$, $x_i$ is missing in the product, so I'd suggest that, as the index counts upwards, the missed out value in the product moves through the $x_i's$, i.e.



$$sum_i=1^nx_1x_2dots x_i-1x_i+1dots x_n= (x_2dots x_n) + (x_1x_3dots x_n)+dots +(x_1dots x_n-1)$$



Or, in a little more compact notation:



$$sum_i=1^nx_1x_2dots x_i-1x_i+1dots x_n=sum_i=1^nprod_jleq n, jneq ix_j=prod_jleq n, jneq 1x_j+dots +prod_jleq n, jneq nx_j$$






share|cite|improve this answer






















  • I know it isn't proper etiquette to respond with non-constructive comments, but cheers! I also need to rethink my life choices...
    – Barycentric_Bash
    Aug 11 at 20:50










  • @Barycentric_Bash To continue "non-proper etiquette", let me ask, why do you need to rethink you life choices?
    – zzuussee
    Aug 11 at 20:51











  • I have to confess that I spent upwards of ~30 min trying to find a 'solution' to my problem because i didn't see the missing xi
    – Barycentric_Bash
    Aug 11 at 20:59






  • 3




    @Barycentric_Bash Notation is the burden all of us have to carry when doing mathematics. The foremost important thing is to understand. So, I think, don't worry to much about that.
    – zzuussee
    Aug 11 at 21:01












up vote
3
down vote



accepted







up vote
3
down vote



accepted






Note, that at every summation step $i$, $x_i$ is missing in the product, so I'd suggest that, as the index counts upwards, the missed out value in the product moves through the $x_i's$, i.e.



$$sum_i=1^nx_1x_2dots x_i-1x_i+1dots x_n= (x_2dots x_n) + (x_1x_3dots x_n)+dots +(x_1dots x_n-1)$$



Or, in a little more compact notation:



$$sum_i=1^nx_1x_2dots x_i-1x_i+1dots x_n=sum_i=1^nprod_jleq n, jneq ix_j=prod_jleq n, jneq 1x_j+dots +prod_jleq n, jneq nx_j$$






share|cite|improve this answer














Note, that at every summation step $i$, $x_i$ is missing in the product, so I'd suggest that, as the index counts upwards, the missed out value in the product moves through the $x_i's$, i.e.



$$sum_i=1^nx_1x_2dots x_i-1x_i+1dots x_n= (x_2dots x_n) + (x_1x_3dots x_n)+dots +(x_1dots x_n-1)$$



Or, in a little more compact notation:



$$sum_i=1^nx_1x_2dots x_i-1x_i+1dots x_n=sum_i=1^nprod_jleq n, jneq ix_j=prod_jleq n, jneq 1x_j+dots +prod_jleq n, jneq nx_j$$







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Aug 11 at 20:50

























answered Aug 11 at 20:42









zzuussee

1,701520




1,701520











  • I know it isn't proper etiquette to respond with non-constructive comments, but cheers! I also need to rethink my life choices...
    – Barycentric_Bash
    Aug 11 at 20:50










  • @Barycentric_Bash To continue "non-proper etiquette", let me ask, why do you need to rethink you life choices?
    – zzuussee
    Aug 11 at 20:51











  • I have to confess that I spent upwards of ~30 min trying to find a 'solution' to my problem because i didn't see the missing xi
    – Barycentric_Bash
    Aug 11 at 20:59






  • 3




    @Barycentric_Bash Notation is the burden all of us have to carry when doing mathematics. The foremost important thing is to understand. So, I think, don't worry to much about that.
    – zzuussee
    Aug 11 at 21:01
















  • I know it isn't proper etiquette to respond with non-constructive comments, but cheers! I also need to rethink my life choices...
    – Barycentric_Bash
    Aug 11 at 20:50










  • @Barycentric_Bash To continue "non-proper etiquette", let me ask, why do you need to rethink you life choices?
    – zzuussee
    Aug 11 at 20:51











  • I have to confess that I spent upwards of ~30 min trying to find a 'solution' to my problem because i didn't see the missing xi
    – Barycentric_Bash
    Aug 11 at 20:59






  • 3




    @Barycentric_Bash Notation is the burden all of us have to carry when doing mathematics. The foremost important thing is to understand. So, I think, don't worry to much about that.
    – zzuussee
    Aug 11 at 21:01















I know it isn't proper etiquette to respond with non-constructive comments, but cheers! I also need to rethink my life choices...
– Barycentric_Bash
Aug 11 at 20:50




I know it isn't proper etiquette to respond with non-constructive comments, but cheers! I also need to rethink my life choices...
– Barycentric_Bash
Aug 11 at 20:50












@Barycentric_Bash To continue "non-proper etiquette", let me ask, why do you need to rethink you life choices?
– zzuussee
Aug 11 at 20:51





@Barycentric_Bash To continue "non-proper etiquette", let me ask, why do you need to rethink you life choices?
– zzuussee
Aug 11 at 20:51













I have to confess that I spent upwards of ~30 min trying to find a 'solution' to my problem because i didn't see the missing xi
– Barycentric_Bash
Aug 11 at 20:59




I have to confess that I spent upwards of ~30 min trying to find a 'solution' to my problem because i didn't see the missing xi
– Barycentric_Bash
Aug 11 at 20:59




3




3




@Barycentric_Bash Notation is the burden all of us have to carry when doing mathematics. The foremost important thing is to understand. So, I think, don't worry to much about that.
– zzuussee
Aug 11 at 21:01




@Barycentric_Bash Notation is the burden all of us have to carry when doing mathematics. The foremost important thing is to understand. So, I think, don't worry to much about that.
– zzuussee
Aug 11 at 21:01












 

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