Can anyone help with find the properties associated with $prod$ of product [closed]

Clash Royale CLAN TAG#URR8PPP

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I can easily find some commonly used properties of $displaystylesum $ but I can't seem to find anything about commonly known formulas of $displaystyleprod$ except $displaystyleprod_i=1^n i = n! $

Thanks! Also do let me know what did you type in the search bar.

products infinite-product

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edited Jul 26 at 18:10

Daniel Buck

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asked Jul 26 at 9:43

William

731214

closed as unclear what you're asking by mathcounterexamples.net, Claudius, Did, Delta-u, Isaac Browne Jul 26 at 22:31

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.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Can you precise what you're looking for? Finite products? Infinite products? General algebraic properties?
  – mathcounterexamples.net
  Jul 26 at 9:48
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @mathcounterexamples.net you know... The basic ones like we have for sigma i and i^2 and i^3 .... The ones that are a "must know"
  – William
  Jul 26 at 9:51
  

  
  
  
  

add a comment | 

up vote
-1
down vote

favorite

I can easily find some commonly used properties of $displaystylesum $ but I can't seem to find anything about commonly known formulas of $displaystyleprod$ except $displaystyleprod_i=1^n i = n! $

Thanks! Also do let me know what did you type in the search bar.

products infinite-product

share|cite|improve this question

edited Jul 26 at 18:10

Daniel Buck

2,3041623

asked Jul 26 at 9:43

William

731214

closed as unclear what you're asking by mathcounterexamples.net, Claudius, Did, Delta-u, Isaac Browne Jul 26 at 22:31

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.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Can you precise what you're looking for? Finite products? Infinite products? General algebraic properties?
  – mathcounterexamples.net
  Jul 26 at 9:48
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @mathcounterexamples.net you know... The basic ones like we have for sigma i and i^2 and i^3 .... The ones that are a "must know"
  – William
  Jul 26 at 9:51
  

  
  
  
  

add a comment | 

up vote
-1
down vote

favorite

up vote
-1
down vote

favorite

I can easily find some commonly used properties of $displaystylesum $ but I can't seem to find anything about commonly known formulas of $displaystyleprod$ except $displaystyleprod_i=1^n i = n! $

Thanks! Also do let me know what did you type in the search bar.

products infinite-product

share|cite|improve this question

edited Jul 26 at 18:10

Daniel Buck

2,3041623

asked Jul 26 at 9:43

William

731214

I can easily find some commonly used properties of $displaystylesum $ but I can't seem to find anything about commonly known formulas of $displaystyleprod$ except $displaystyleprod_i=1^n i = n! $

Thanks! Also do let me know what did you type in the search bar.

products infinite-product

share|cite|improve this question

edited Jul 26 at 18:10

Daniel Buck

2,3041623

asked Jul 26 at 9:43

William

731214

share|cite|improve this question

share|cite|improve this question

edited Jul 26 at 18:10

Daniel Buck

2,3041623

edited Jul 26 at 18:10

Daniel Buck

2,3041623

edited Jul 26 at 18:10

Daniel Buck

2,3041623

2,3041623

asked Jul 26 at 9:43

William

731214

asked Jul 26 at 9:43

William

731214

asked Jul 26 at 9:43

William

731214

731214

closed as unclear what you're asking by mathcounterexamples.net, Claudius, Did, Delta-u, Isaac Browne Jul 26 at 22:31

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 mathcounterexamples.net, Claudius, Did, Delta-u, Isaac Browne Jul 26 at 22:31

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.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Can you precise what you're looking for? Finite products? Infinite products? General algebraic properties?
  – mathcounterexamples.net
  Jul 26 at 9:48
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @mathcounterexamples.net you know... The basic ones like we have for sigma i and i^2 and i^3 .... The ones that are a "must know"
  – William
  Jul 26 at 9:51
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Can you precise what you're looking for? Finite products? Infinite products? General algebraic properties?
  – mathcounterexamples.net
  Jul 26 at 9:48
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @mathcounterexamples.net you know... The basic ones like we have for sigma i and i^2 and i^3 .... The ones that are a "must know"
  – William
  Jul 26 at 9:51
  

  
  
  
  

Can you precise what you're looking for? Finite products? Infinite products? General algebraic properties?
– mathcounterexamples.net
Jul 26 at 9:48

Can you precise what you're looking for? Finite products? Infinite products? General algebraic properties?
– mathcounterexamples.net
Jul 26 at 9:48

@mathcounterexamples.net you know... The basic ones like we have for sigma i and i^2 and i^3 .... The ones that are a "must know"
– William
Jul 26 at 9:51

@mathcounterexamples.net you know... The basic ones like we have for sigma i and i^2 and i^3 .... The ones that are a "must know"
– William
Jul 26 at 9:51

add a comment | 

3 Answers
3

active

oldest

votes

up vote
1
down vote

$$
logprod ?=sumlog,?
$$
allows you to transfer properties of summations to products.

share|cite|improve this answer

edited Jul 26 at 17:47

answered Jul 26 at 12:59

Yves Daoust

111k665203

add a comment | 

up vote
1
down vote

Examples: The Wallis Product for $pi.$ And Euler's product $sinpi x=pi x prod_n=1^infty(1-x^2/n^2).$ Neither of these is easy to prove. See "Wallis product" in Wikipedia.

A useful theorem, which you can find (for example) in the old classic Infinite Sequences And Series, by Bromwich:

(1). If $a_ngeq 0$ for every $n$ then $sum_n=1^inftya_n$ converges iff $prod_n=1^infty(1+a_n)$ converges.

(2). If $1>a_ngeq 0$ for every $n$ then $sum_n=1^inftya_n$ converges iff $prod_n=1^infty(1-a_n)>0.$

Euler used (2) to show that $sum_pin P;(1/p)=infty,$ where $P$ is the set of primes (and an immediate corollary, a proof that $P$ is infinite, obtained by analytic methods), as follows:

For $2leq Min Bbb N$ let $P(M)$ be the set of primes that are not more than $M.$ We have $$prod_pin P(1-1/p)^-1geq prod_pin P(M)(1-1/p)^-1=$$ $$=prod_pin P(M)(sum_j=0^inftyp^-j)>$$ $$>prod_pin P(M)(sum_j=0^Mp^-j).$$ Now if the last expression above is completely expanded, every $n^-1$ for $1leq nleq M$ will appear at least once as a term (Exactly once, actually, because of unique prime decomposition). There will generally be a whole lot of other terms as well. So for every $Mgeq 2 $ we have $$prod_pin P(1-1/p)^-1>sum_n=1^Mn^-1.$$ But since $sum_n=1^inftyn^-1$ diverges we must have $prod_pin P;(1-1/p)^-1=infty, $ so $prod_pin P;(1-1/p)=0.$ By (2), therefore $sum_pin P;(1/p) =infty.$

share|cite|improve this answer

edited Jul 26 at 20:03

answered Jul 26 at 19:52

DanielWainfleet

31.5k31542

• 
  
  
  

  
  

  
  
  
  
  
  

  
  The Bromwich book can be read without any pre-requisites.
  – DanielWainfleet
  Jul 26 at 20:00
  

  
  
  
  

add a comment | 

up vote
0
down vote

One key word is "finite products". Some hints are given in introductory combinatorics books, e.g. in the following books.

Riordan, J.: An Introduction to Combinatorial Analysis. Princeton University Press, 1958

Comtet, L.: Advanced Combinatorics: The Art of Finite and Innite Expansions.
Reidel, 1974

Graham, R. L.; Knuth, D. E.; Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley, 1994

Charalambides, Ch. A.: Enumerative combinatorics. CRC Press, 2002

share|cite|improve this answer

answered Jul 26 at 12:39

IV_

940221

add a comment | 

3 Answers
3

active

oldest

votes

3 Answers
3

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
1
down vote

$$
logprod ?=sumlog,?
$$
allows you to transfer properties of summations to products.

share|cite|improve this answer

edited Jul 26 at 17:47

answered Jul 26 at 12:59

Yves Daoust

111k665203

add a comment | 

up vote
1
down vote

$$
logprod ?=sumlog,?
$$
allows you to transfer properties of summations to products.

share|cite|improve this answer

edited Jul 26 at 17:47

answered Jul 26 at 12:59

Yves Daoust

111k665203

add a comment | 

up vote
1
down vote

up vote
1
down vote

$$
logprod ?=sumlog,?
$$
allows you to transfer properties of summations to products.

share|cite|improve this answer

edited Jul 26 at 17:47

answered Jul 26 at 12:59

Yves Daoust

111k665203

$$
logprod ?=sumlog,?
$$
allows you to transfer properties of summations to products.

share|cite|improve this answer

edited Jul 26 at 17:47

answered Jul 26 at 12:59

Yves Daoust

111k665203

share|cite|improve this answer

share|cite|improve this answer

edited Jul 26 at 17:47

edited Jul 26 at 17:47

edited Jul 26 at 17:47

answered Jul 26 at 12:59

Yves Daoust

111k665203

answered Jul 26 at 12:59

Yves Daoust

111k665203

answered Jul 26 at 12:59

Yves Daoust

111k665203

111k665203

add a comment | 

add a comment | 

up vote
1
down vote

Examples: The Wallis Product for $pi.$ And Euler's product $sinpi x=pi x prod_n=1^infty(1-x^2/n^2).$ Neither of these is easy to prove. See "Wallis product" in Wikipedia.

A useful theorem, which you can find (for example) in the old classic Infinite Sequences And Series, by Bromwich:

(1). If $a_ngeq 0$ for every $n$ then $sum_n=1^inftya_n$ converges iff $prod_n=1^infty(1+a_n)$ converges.

(2). If $1>a_ngeq 0$ for every $n$ then $sum_n=1^inftya_n$ converges iff $prod_n=1^infty(1-a_n)>0.$

Euler used (2) to show that $sum_pin P;(1/p)=infty,$ where $P$ is the set of primes (and an immediate corollary, a proof that $P$ is infinite, obtained by analytic methods), as follows:

For $2leq Min Bbb N$ let $P(M)$ be the set of primes that are not more than $M.$ We have $$prod_pin P(1-1/p)^-1geq prod_pin P(M)(1-1/p)^-1=$$ $$=prod_pin P(M)(sum_j=0^inftyp^-j)>$$ $$>prod_pin P(M)(sum_j=0^Mp^-j).$$ Now if the last expression above is completely expanded, every $n^-1$ for $1leq nleq M$ will appear at least once as a term (Exactly once, actually, because of unique prime decomposition). There will generally be a whole lot of other terms as well. So for every $Mgeq 2 $ we have $$prod_pin P(1-1/p)^-1>sum_n=1^Mn^-1.$$ But since $sum_n=1^inftyn^-1$ diverges we must have $prod_pin P;(1-1/p)^-1=infty, $ so $prod_pin P;(1-1/p)=0.$ By (2), therefore $sum_pin P;(1/p) =infty.$

share|cite|improve this answer

edited Jul 26 at 20:03

answered Jul 26 at 19:52

DanielWainfleet

31.5k31542

• 
  
  
  

  
  

  
  
  
  
  
  

  
  The Bromwich book can be read without any pre-requisites.
  – DanielWainfleet
  Jul 26 at 20:00
  

  
  
  
  

add a comment | 

up vote
1
down vote

Examples: The Wallis Product for $pi.$ And Euler's product $sinpi x=pi x prod_n=1^infty(1-x^2/n^2).$ Neither of these is easy to prove. See "Wallis product" in Wikipedia.

A useful theorem, which you can find (for example) in the old classic Infinite Sequences And Series, by Bromwich:

(1). If $a_ngeq 0$ for every $n$ then $sum_n=1^inftya_n$ converges iff $prod_n=1^infty(1+a_n)$ converges.

(2). If $1>a_ngeq 0$ for every $n$ then $sum_n=1^inftya_n$ converges iff $prod_n=1^infty(1-a_n)>0.$

Euler used (2) to show that $sum_pin P;(1/p)=infty,$ where $P$ is the set of primes (and an immediate corollary, a proof that $P$ is infinite, obtained by analytic methods), as follows:

For $2leq Min Bbb N$ let $P(M)$ be the set of primes that are not more than $M.$ We have $$prod_pin P(1-1/p)^-1geq prod_pin P(M)(1-1/p)^-1=$$ $$=prod_pin P(M)(sum_j=0^inftyp^-j)>$$ $$>prod_pin P(M)(sum_j=0^Mp^-j).$$ Now if the last expression above is completely expanded, every $n^-1$ for $1leq nleq M$ will appear at least once as a term (Exactly once, actually, because of unique prime decomposition). There will generally be a whole lot of other terms as well. So for every $Mgeq 2 $ we have $$prod_pin P(1-1/p)^-1>sum_n=1^Mn^-1.$$ But since $sum_n=1^inftyn^-1$ diverges we must have $prod_pin P;(1-1/p)^-1=infty, $ so $prod_pin P;(1-1/p)=0.$ By (2), therefore $sum_pin P;(1/p) =infty.$

share|cite|improve this answer

edited Jul 26 at 20:03

answered Jul 26 at 19:52

DanielWainfleet

31.5k31542

• 
  
  
  

  
  

  
  
  
  
  
  

  
  The Bromwich book can be read without any pre-requisites.
  – DanielWainfleet
  Jul 26 at 20:00
  

  
  
  
  

add a comment | 

up vote
1
down vote

up vote
1
down vote

Examples: The Wallis Product for $pi.$ And Euler's product $sinpi x=pi x prod_n=1^infty(1-x^2/n^2).$ Neither of these is easy to prove. See "Wallis product" in Wikipedia.

A useful theorem, which you can find (for example) in the old classic Infinite Sequences And Series, by Bromwich:

(1). If $a_ngeq 0$ for every $n$ then $sum_n=1^inftya_n$ converges iff $prod_n=1^infty(1+a_n)$ converges.

(2). If $1>a_ngeq 0$ for every $n$ then $sum_n=1^inftya_n$ converges iff $prod_n=1^infty(1-a_n)>0.$

Euler used (2) to show that $sum_pin P;(1/p)=infty,$ where $P$ is the set of primes (and an immediate corollary, a proof that $P$ is infinite, obtained by analytic methods), as follows:

For $2leq Min Bbb N$ let $P(M)$ be the set of primes that are not more than $M.$ We have $$prod_pin P(1-1/p)^-1geq prod_pin P(M)(1-1/p)^-1=$$ $$=prod_pin P(M)(sum_j=0^inftyp^-j)>$$ $$>prod_pin P(M)(sum_j=0^Mp^-j).$$ Now if the last expression above is completely expanded, every $n^-1$ for $1leq nleq M$ will appear at least once as a term (Exactly once, actually, because of unique prime decomposition). There will generally be a whole lot of other terms as well. So for every $Mgeq 2 $ we have $$prod_pin P(1-1/p)^-1>sum_n=1^Mn^-1.$$ But since $sum_n=1^inftyn^-1$ diverges we must have $prod_pin P;(1-1/p)^-1=infty, $ so $prod_pin P;(1-1/p)=0.$ By (2), therefore $sum_pin P;(1/p) =infty.$

share|cite|improve this answer

edited Jul 26 at 20:03

answered Jul 26 at 19:52

DanielWainfleet

31.5k31542

Examples: The Wallis Product for $pi.$ And Euler's product $sinpi x=pi x prod_n=1^infty(1-x^2/n^2).$ Neither of these is easy to prove. See "Wallis product" in Wikipedia.

A useful theorem, which you can find (for example) in the old classic Infinite Sequences And Series, by Bromwich:

(1). If $a_ngeq 0$ for every $n$ then $sum_n=1^inftya_n$ converges iff $prod_n=1^infty(1+a_n)$ converges.

(2). If $1>a_ngeq 0$ for every $n$ then $sum_n=1^inftya_n$ converges iff $prod_n=1^infty(1-a_n)>0.$

Euler used (2) to show that $sum_pin P;(1/p)=infty,$ where $P$ is the set of primes (and an immediate corollary, a proof that $P$ is infinite, obtained by analytic methods), as follows:

For $2leq Min Bbb N$ let $P(M)$ be the set of primes that are not more than $M.$ We have $$prod_pin P(1-1/p)^-1geq prod_pin P(M)(1-1/p)^-1=$$ $$=prod_pin P(M)(sum_j=0^inftyp^-j)>$$ $$>prod_pin P(M)(sum_j=0^Mp^-j).$$ Now if the last expression above is completely expanded, every $n^-1$ for $1leq nleq M$ will appear at least once as a term (Exactly once, actually, because of unique prime decomposition). There will generally be a whole lot of other terms as well. So for every $Mgeq 2 $ we have $$prod_pin P(1-1/p)^-1>sum_n=1^Mn^-1.$$ But since $sum_n=1^inftyn^-1$ diverges we must have $prod_pin P;(1-1/p)^-1=infty, $ so $prod_pin P;(1-1/p)=0.$ By (2), therefore $sum_pin P;(1/p) =infty.$

share|cite|improve this answer

edited Jul 26 at 20:03

answered Jul 26 at 19:52

DanielWainfleet

31.5k31542

share|cite|improve this answer

share|cite|improve this answer

edited Jul 26 at 20:03

edited Jul 26 at 20:03

edited Jul 26 at 20:03

answered Jul 26 at 19:52

DanielWainfleet

31.5k31542

answered Jul 26 at 19:52

DanielWainfleet

31.5k31542

answered Jul 26 at 19:52

DanielWainfleet

31.5k31542

31.5k31542

• 
  
  
  

  
  

  
  
  
  
  
  

  
  The Bromwich book can be read without any pre-requisites.
  – DanielWainfleet
  Jul 26 at 20:00
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  The Bromwich book can be read without any pre-requisites.
  – DanielWainfleet
  Jul 26 at 20:00
  

  
  
  
  

The Bromwich book can be read without any pre-requisites.
– DanielWainfleet
Jul 26 at 20:00

The Bromwich book can be read without any pre-requisites.
– DanielWainfleet
Jul 26 at 20:00

add a comment | 

up vote
0
down vote

One key word is "finite products". Some hints are given in introductory combinatorics books, e.g. in the following books.

Riordan, J.: An Introduction to Combinatorial Analysis. Princeton University Press, 1958

Comtet, L.: Advanced Combinatorics: The Art of Finite and Innite Expansions.
Reidel, 1974

Graham, R. L.; Knuth, D. E.; Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley, 1994

Charalambides, Ch. A.: Enumerative combinatorics. CRC Press, 2002

share|cite|improve this answer

answered Jul 26 at 12:39

IV_

940221

add a comment | 

up vote
0
down vote

One key word is "finite products". Some hints are given in introductory combinatorics books, e.g. in the following books.

Riordan, J.: An Introduction to Combinatorial Analysis. Princeton University Press, 1958

Comtet, L.: Advanced Combinatorics: The Art of Finite and Innite Expansions.
Reidel, 1974

Graham, R. L.; Knuth, D. E.; Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley, 1994

Charalambides, Ch. A.: Enumerative combinatorics. CRC Press, 2002

share|cite|improve this answer

answered Jul 26 at 12:39

IV_

940221

add a comment | 

up vote
0
down vote

up vote
0
down vote

One key word is "finite products". Some hints are given in introductory combinatorics books, e.g. in the following books.

Riordan, J.: An Introduction to Combinatorial Analysis. Princeton University Press, 1958

Comtet, L.: Advanced Combinatorics: The Art of Finite and Innite Expansions.
Reidel, 1974

Graham, R. L.; Knuth, D. E.; Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley, 1994

Charalambides, Ch. A.: Enumerative combinatorics. CRC Press, 2002

share|cite|improve this answer

answered Jul 26 at 12:39

IV_

940221

One key word is "finite products". Some hints are given in introductory combinatorics books, e.g. in the following books.

Riordan, J.: An Introduction to Combinatorial Analysis. Princeton University Press, 1958

Comtet, L.: Advanced Combinatorics: The Art of Finite and Innite Expansions.
Reidel, 1974

Graham, R. L.; Knuth, D. E.; Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley, 1994

Charalambides, Ch. A.: Enumerative combinatorics. CRC Press, 2002

share|cite|improve this answer

answered Jul 26 at 12:39

IV_

940221

share|cite|improve this answer

share|cite|improve this answer

answered Jul 26 at 12:39

IV_

940221

answered Jul 26 at 12:39

IV_

940221

answered Jul 26 at 12:39

IV_

940221

940221

add a comment | 

add a comment |