Sums to Products and Products to Sums

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In summary, the conversation discusses the conversion of infinite series to infinite products and vice-versa. It provides a formula for converting a series to a product using telescoping sums and simplifying the expression. The process is then reversed to convert a product to a series. The source for this technique is "Theory and Applications of Infinite Series" by K. Knopp.
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benorin
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This discussion is that of converting infinite series to infinite products and vice-versa in hopes of, say, ending the shortage of infinite product tables.

Suppose the given series is

[tex]\sum_{k=0}^{\infty} a_k[/tex]

Let [itex]S_n[/tex] denote the nth partial sum, viz.

[tex]S_n=\sum_{k=0}^{n} a_k[/tex]

so that, if [itex]S_{n}\neq 0,\forall n\in\mathbb{N}[/itex] , then

[tex]S_n=S_{0} \frac{S_{1}}{S_{0}} \frac{S_{2}}{S_{1}} \cdot\cdot\cdot \frac{S_{n}}{S_{n-1}} = S_{0} \prod_{k=1}^{n} \frac{S_{k}}{S_{k-1}}[/tex]

which is a pretty basic telescoping product, and it will simplify upon noticing that [itex]S_{k}= a_{k} + S_{k-1}[/itex], and that [itex]S_{0}= a_{0}[/itex], whence

[tex]S_n= S_{0} \prod_{k=1}^{n} \frac{S_{k}}{S_{k-1}} = a_{0} \prod_{k=1}^{n} \left( 1+ \frac{a_{k}}{S_{k-1}} \right) = a_{0} \prod_{k=1}^{n} \left( 1+ \frac{a_{k}}{a_{0}+a_{1}+\cdot\cdot\cdot + a_{k-1}} \right) [/tex]

and hence, taking the limit as [itex] n\rightarrow \infty[/itex], we have

[tex]\sum_{k=0}^{\infty} a_k = a_{0} \prod_{k=1}^{\infty} \left( 1+ \frac{a_{k}}{a_{0}+a_{1}+\cdot\cdot\cdot + a_{k-1}} \right) [/tex]

now you can convert an infinite series to an infinite product.

So the vice-versa part goes like this:

Suppose the given product is

[tex]\prod_{k=0}^{\infty} a_k[/tex]

Let [itex]\rho _n[/tex] denote the nth partial product, viz.

[tex]\rho_{n}=\prod_{k=0}^{n} a_k[/tex]

so that, if [itex]\rho_{n}\neq 0,\forall n\in\mathbb{N}[/itex] , then

[tex]\rho_{n} = \rho_{0} + \left( \rho_{1} - \rho_{0} \right) + \left( \rho_{2} - \rho_{1} \right) + \cdot\cdot\cdot + \left( \rho_{n} - \rho_{n-1} \right) = \rho_{0} + \sum_{k=1}^{n} \left( \rho_{k} - \rho_{k-1} \right) [/tex]

which is an extemely basic telescoping sum, and it will simplify upon noticing that [itex]\rho_{k}= a_{k} \rho_{k-1}[/itex], and that [itex] \rho_{0}= a_{0}[/itex], whence

[tex]\rho_{n} = \rho_{0} + \sum_{k=1}^{n} \left( \rho_{k} - \rho_{k-1} \right) = a_{0} + \sum_{k=1}^{n} \rho_{k-1} \left( a_{k} - 1 \right) = a_{0} + \sum_{k=1}^{n} a_{0}a_{1}\cdot\cdot\cdot a_{k-1} \left( a_{k} - 1 \right) [/tex]

and hence, taking the limit as [itex] n\rightarrow \infty[/itex], we have

[tex]\prod_{k=0}^{\infty} a_k = a_{0} + \sum_{k=1}^{n} a_{0}a_{1}\cdot\cdot\cdot a_{k-1} \left( a_{k} - 1 \right) [/tex]

and now you can convert an infinite product to an infinite series.

So, go on, have fun with it...

P.S. I swipped this technique from Theroy and Applications of Infinite Series by K. Knopp :wink: a very excellent text.
 
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  • #2
Oops, typo: that last tex line should read

[tex]\prod_{k=0}^{\infty} a_k = a_{0} + \sum_{k=1}^{\infty} a_{0}a_{1}\cdot\cdot\cdot a_{k-1} \left( a_{k} - 1 \right) [/tex]
 
  • #3
This is great work. Can someone prove this though or provide a link to a source please, though?
 
  • #4
Your edit answers my question, thanks. "Theory and Applications of Infinite Series" by K. Knoppz.
 

What is the concept of "Sums to Products and Products to Sums" in mathematics?

"Sums to Products and Products to Sums" refers to a mathematical technique used to rewrite algebraic expressions containing sums or products into a different form. This technique is helpful in simplifying expressions, solving equations, and proving identities.

How do you convert a sum to a product using the "Sums to Products" technique?

To convert a sum to a product, you can use the following formula: (a+b)(a+c) = a2+ab+ac+bc. This formula allows you to expand a sum of two terms into a product of four terms. You can continue this process with more terms to convert a larger sum into a product.

How do you convert a product to a sum using the "Products to Sums" technique?

To convert a product to a sum, you can use the following formula: ab = (a+b)2 - (a2 + b2). This formula allows you to expand a product of two terms into a sum of three terms. You can continue this process with more terms to convert a larger product into a sum.

What are some real-life applications of "Sums to Products and Products to Sums" in mathematics?

The "Sums to Products and Products to Sums" technique is used in various fields of mathematics, including algebra, trigonometry, and calculus. It is also commonly used in physics and engineering to simplify equations and solve problems involving sums and products.

Are there any limitations to using the "Sums to Products and Products to Sums" technique?

While the "Sums to Products and Products to Sums" technique is a useful tool in mathematics, it may not always be applicable or efficient in solving certain problems. It is important to understand when and how to use this technique appropriately.

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