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# infinite product of differences $1\!-\!a_{i}$

We consider the infinite products of the form

$\displaystyle\prod_{{i=1}}^{\infty}(1\!-\!a_{i})\,=\,(1\!-\!a_{1})(1\!-\!a_{2}% )(1\!-\!a_{3})\cdots$ | (1) |

and the series $a_{1}\!+\!a_{2}\!+\!a_{3}\!+\ldots$ where the numbers $a_{i}$ are nonnegative reals.

Example. $(1\!-\!\frac{1}{2})(1\!-\!\frac{1}{3})(1\!-\!\frac{1}{4})\cdots\;=\;0$; see the harmonic series.

Proof. $1^{\circ}$. Now we have $\displaystyle\lim_{{i\to\infty}}a_{i}=0$ (see the necessary condition of convergence of series), and so $a_{i}<\frac{1}{2}$ when $i\geqq i_{0}$. We write

$\displaystyle\prod_{{i=1}}^{\infty}(1\!-\!a_{i})\,=\,\prod_{{i=1}}^{{i_{0}-1}}% (1\!-\!a_{i})\prod_{{i=i_{0}}}^{\infty}(1\!-\!a_{i})$ | (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

$\displaystyle\prod_{{i=i_{0}}}^{n}(1\!-\!a_{i})\,=\,\frac{1}{\prod_{{i=i_{0}}}% ^{n}\left(1+\frac{a_{i}}{1\!-\!a_{i}}\right)}.$ | (3) |

As $a_{i}<\frac{1}{2}$, we have $\displaystyle\frac{1}{1\!-\!a_{i}}<2$ and thus
$\displaystyle 0<\frac{a_{i}}{1\!-\!a_{i}}<2\cdot a_{i}$, and therefore the series
$\displaystyle\sum_{{i=i_{0}}}^{\infty}\frac{a_{i}}{1\!-\!a_{i}}$ with nonnegative terms is absolutely convergent. The theorem of the parent entry then says that the product in the denominator of the right hand side of (3) tends, as $n\to\infty$, 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 independent on the order and it vanishes only along with some of its factors.

$2^{\circ}$. There is an $i_{0}$ such that $a_{i}<1$ when $i\geqq i_{0}$, whence $\frac{a_{i}}{1\!-\!a_{i}}>a_{i}$ and the series $\displaystyle\sum_{{i=i_{0}}}^{\infty}\frac{a_{i}}{1\!-\!a_{i}}$ diverges. The denominator of the right hand side of (3) tends, as $n\to\infty$, 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).

## Mathematics Subject Classification

40A20*no label found*26E99

*no label found*

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