Sequences and Series/Infinite products
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Definition (infinite product):
Let be a sequence of numbers in or . If the limit
exists, it is called the infinite product of and denoted by
- .
Proposition (necessary condition for convergence of infinite products):
In order for the infinite product
of a sequence to exist and not to be zero, it is necessary that
- .
Proof: Suppose that not . Then there exists and an infinite sequence such that for all we have . Thus, upon denoting
- ,
we will have
- .
Suppose for a contradiction that exited and was equal to . Then when is sufficiently large, we will have
- ,
which is a contradiction.
Proposition (series criterion for the convergence of infinite products):
Let be a sequence of real numbers. If
- ,
then
converges.
Proof: