Find all real numbers such that
The 'infinite product' ...
(1)
... diverges if the corresponding 'infinite sum'...
(2)
... diverges. In this case the 'infinite sum' is...
(3)
... and we can use the 'integral test' to verify its convergence. If we have...
(4)
... and the integral from to diverges. If we have...
(5)
... and the integral from to converges. The conclusion is that the 'infinite product'...
(6)
... converges for and diverges for ...
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A precise enough definition of 'infinite product' is given here...
Infinite product - Wikipedia, the free encyclopedia
If You have an infinite product in the form...
(1)
... it is said to be convergent if the limit exists and it is not zero. In all other cases it is said to be divergent. If we can write...
(2)
... so that the convergence of the infinite product is equivalent of the convergence of the 'infinite sum' of logarithms. If a product of infinite positive terms diverges the limit can be or . An useful criterion of convergence of an infinite product is allowable when the general term can be written as . In this case the (1) converges if converges the 'infinite sum'...
(3)
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