infinite product of differences
We consider the infinite products of the form
and the series where the numbers are nonnegative reals.
If but the series diverges, then the value of the infinite product is always zero though no of the factors were zero.
Example. ; see the harmonic series.
Proof. . Now we have (see the necessary condition of convergence of series), and so when . We write
and set in the last product
As , we have and thus
, and therefore the series
with nonnegative is absolutely convergent. The theorem of the http://planetmath.org/node/6204parent entry then says that the product in the denominator of the right hand side of (3) tends, as , to a finite non-zero limit, which don’t depend on the order of the factors. Consequently, the same concerns the product of the left hand side of (3). By (2), we now infer that the given product (1) converges, its value is on the order and it vanishes only along with some of its factors.
. There is an such that when , whence and the series diverges. The denominator of the right hand side of (3) tends, as , to the infinity and thus the product of the left hand side to 0. Hence the value of (1) is necessarily 0, also when all factors were distinct from 0.
- 1 E. Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset III.2. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1940).
|Title||infinite product of differences|
|Date of creation||2013-03-22 18:39:45|
|Last modified on||2013-03-22 18:39:45|
|Last modified by||pahio (2872)|