infinite product of differences $1\text{tmspace}.1667em\text{tmspace}.1667em{a}_{i}$
We consider the infinite products of the form
$\prod _{i=1}^{\mathrm{\infty}}}(1{a}_{i})=(1{a}_{1})(1{a}_{2})(1{a}_{3})\mathrm{\cdots$  (1) 
and the series ${a}_{1}+{a}_{2}+{a}_{3}+\mathrm{\dots}$ where the numbers ${a}_{i}$ are nonnegative reals.
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If $\underset{i\to \mathrm{\infty}}{lim}{a}_{i}=0$ but the series diverges, then the value of the infinite product is always zero though no of the factors were zero.
Example. $(1\frac{1}{2})(1\frac{1}{3})(1\frac{1}{4})\mathrm{\cdots}=\mathrm{\hspace{0.33em}0}$; see the harmonic series^{}.
Proof. ${1}^{\circ}$. Now we have $\underset{i\to \mathrm{\infty}}{lim}{a}_{i}=0$ (see the necessary condition of convergence of series), and so $$ when $i\geqq {i}_{0}$. We write
$\prod _{i=1}^{\mathrm{\infty}}}(1{a}_{i})={\displaystyle \prod _{i=1}^{{i}_{0}1}}(1{a}_{i}){\displaystyle \prod _{i={i}_{0}}^{\mathrm{\infty}}}(1{a}_{i})$  (2) 
and set in the last product
$$1{a}_{i}=\frac{1}{\frac{1}{1{a}_{i}}}=\frac{1}{1+\frac{{a}_{i}}{1{a}_{i}}},$$ 
whence
$\prod _{i={i}_{0}}^{n}}(1{a}_{i})={\displaystyle \frac{1}{{\prod}_{i={i}_{0}}^{n}\left(1+\frac{{a}_{i}}{1{a}_{i}}\right)}}.$  (3) 
As $$, we have $$ and thus
$$, and therefore the series
$\sum _{i={i}_{0}}^{\mathrm{\infty}}}{\displaystyle \frac{{a}_{i}}{1{a}_{i}}$ 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 $n\to \mathrm{\infty}$, to a finite nonzero 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.
${2}^{\circ}$. There is an ${i}_{0}$ such that $$ when $i\geqq {i}_{0}$, whence $\frac{{a}_{i}}{1{a}_{i}}>{a}_{i}$ and the series $\sum _{i={i}_{0}}^{\mathrm{\infty}}}{\displaystyle \frac{{a}_{i}}{1{a}_{i}}$ diverges. The denominator of the right hand side of (3) tends, as $n\to \mathrm{\infty}$, 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.
References
 1 E. Lindelöf: Differentiali ja integralilasku ja sen sovellutukset III.2. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1940).
Title  infinite product of differences $1\text{tmspace}.1667em\text{tmspace}.1667em{a}_{i}$ 

Canonical name  InfiniteProductOfDifferences1ai 
Date of creation  20130322 18:39:45 
Last modified on  20130322 18:39:45 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  12 
Author  pahio (2872) 
Entry type  Theorem 
Classification  msc 40A20 
Classification  msc 26E99 
Related topic  HarmonicSeriesOfPrimes 
Related topic  InfiniteProductOfSums1A_I 
Related topic  InfiniteProductOfSums1a_i 