Is this product convergent

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  • #1
tpm
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Following Euler if we define the product:

[tex] (x-2^{-s})(x-3^{-s}) (x-5^{-s})(x-7^{-s})...=f(x) [/tex]

taken over all primes and s > 1 ,what would be the value of f(x) ?? i believe that [tex] f(x,s)=1/Li_{s} (x) [/tex] (inverse of Polylogarithm) however I'm not 100 % sure, although for x=1 you get the inverse of Riemann Zeta
 
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  • #2
1. Inverse and Reciprocals aren't the same thing.

2. For any value of s, and x > 1, the terms do not approach 1, but x. x being more than 1, the terms are not approaching 1, so the product does not converge.
 
  • #3
function

Based on the information provided, it is difficult to determine if this product is convergent. Euler's definition of the product does not explicitly state if it is convergent or not. Additionally, the value of f(x) cannot be determined without knowing the specific values of x and s.

However, based on the given information, it seems that the product may have a value of 1/Li_s(x), which is the inverse of the Polylogarithm function. This function is closely related to the Riemann Zeta function, which has been extensively studied in mathematics and has a value of 1 when x=1. Therefore, if x=1, then the product may have a value of 1, which is the inverse of the Riemann Zeta function.

In order to determine if this product is convergent, more information is needed such as the specific values of x and s, and if the product has a finite limit as s approaches infinity. Additionally, further analysis and mathematical techniques may be necessary to determine the convergence of this product.
 

1. Is this product convergent?

The answer to this question depends on the specific product in question. In general, a product is considered convergent if its terms approach a finite value as the number of terms increases. This can be determined by evaluating the limit of the product. If the limit exists and is a finite value, the product is convergent. Otherwise, it is divergent.

2. How do I determine if a product is convergent?

To determine if a product is convergent, you can use various convergence tests such as the ratio test, comparison test, or the integral test. These tests involve evaluating the limit of the product or comparing it to a known convergent or divergent series. If the limit exists and is a finite value, the product is convergent. Otherwise, it is divergent.

3. Can a product be both convergent and divergent?

No, a product cannot be both convergent and divergent. It is either one or the other. If the limit of the product exists and is a finite value, it is convergent. If the limit does not exist or is infinite, it is divergent.

4. Can a product be convergent if its individual terms are divergent?

Yes, a product can still be convergent even if its individual terms are divergent. This is because the behavior of a series is determined by the relationship between the terms, not the individual terms themselves. As long as the limit of the product exists and is a finite value, the product is convergent.

5. How does the convergence of a product affect its sum?

The convergence of a product does not necessarily determine the convergence of its sum. A product can be convergent but its sum can still be divergent, and vice versa. This is because the sum of a product depends on the relationship between the individual terms, not just their convergence or divergence. It is important to evaluate the sum of a product separately from its convergence.

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