proof of convergence criterion for infinite product
Consider the partial product .
By definition we say that the infinite product is convergent iff is convergent.
Since , the infinite product converges to a positive value iff the series is convergent.
In particular, if the infinite product converges to a positive value, then .
, is equivalent to saying
For the second part of the theorem:
converges absolutely to a positive value iff converges absolutely.
as we have seen,
consider: (this is easy to prove since by Taylor’s expansion )
Since we can say that and by the limit comparison test, either both and converge or diverge.
|Title||proof of convergence criterion for infinite product|
|Date of creation||2013-03-22 15:35:36|
|Last modified on||2013-03-22 15:35:36|
|Last modified by||cvalente (11260)|