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Question: How to prove this sum is equal to product ?

Question:How to prove this sum is equal to product ?

janhardo 375 Maple
product summation
January 06 2022
2

Try to prove those two expressions : a sum and product 
First to know if they are correct defined for some values ?

Eulers productformule in Maple

 
> 

 

Eerst de Euler identiteit in Maple opschrijven
via info over series

 

p-Series

sum(1/n^p, n = 1 .. infinity)

Absolute convergence for p > 1

Diverges for p <= 1 

 

 

> 

restart;

sum(1/n^p, n = 1 .. infinity) = product(1/(1-p^(-s)), i = k .. p)

> 

product(f,i=k..n);

> 

product(1/(1 - p^(-s)), i = 1 .. p);

(1-p^(-s))*(-1/(-1+p^(-s)))^(p+1)

(1.1)
> 

sum(1/n^p, n = 1 .. infinity);

sum(1/n^p, n = 1 .. infinity)

(1.2)
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sum(1/n^p, n = 1 .. 4);

1+1/2^p+1/3^p+1/4^p

(1.3)
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product(1/(1 - p^(-s)), i = 1 .. 4);
            

1/(1-p^(-s))^4

(1.4)
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Product(1/(1 - p^(-s)), i = 1 .. 4)= product(1/(1 - p^(-s)), i = 1 .. 4);

Product(1/(1-p^(-s)), i = 1 .. 4) = 1/(1-p^(-s))^4

(1.5)
> 

sum(1/n^p, n = 1 .. infinity) = product(1/(1 - p^(-s)), i = k .. p);

sum(1/n^p, n = 1 .. infinity) = (-1/(-1+p^(-s)))^(p+1)/(-1/(-1+p^(-s)))^k

(1.6)
> 

 

 How to prove this sum(1/n^p, n = 1 .. infinity) = product(1/(1-p^(-s)), i = k .. p)

For the sum i got some terms , but for the product i don't see factors as i use the commands  

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