Let $x_0, x_1, x_2, \ldots \in [0, 1)$ each be a D993: Real number.
Then
\begin{equation}
\prod_{n = 0}^{\infty} (1 - x_n) > 0
\quad \iff \quad
\sum_{n = 0}^{\infty} x_n < \infty
\end{equation}
Result R5166
on D993: Real number
Subresult of R4692: Series diverges for real numbers in the left-closed unit interval iff infinite product of duals vanishes
Subresult of R4692: Series diverges for real numbers in the left-closed unit interval iff infinite product of duals vanishes
Series converges for real numbers in the left-closed unit interval iff infinite product of duals does not vanish
Formulation 1
Let $x_0, x_1, x_2, \ldots \in [0, 1)$ each be a D993: Real number.
Then
\begin{equation}
(1 - x_0) (1 - x_1) (1 - x_2) (1 - x_3) \cdots > 0
\quad \iff \quad
x_0 + x_1 + x_2 + x_3 + \cdots < \infty
\end{equation}
Formulation 2
Let $x_0, x_1, x_2, \ldots \in [0, 1)$ each be a D993: Real number.
Then
\begin{equation}
\lim_{N \to \infty} \prod_{n = 0}^N (1 - x_n) > 0
\quad \iff \quad
\lim_{N \to \infty} \sum_{n = 0}^N x_n < \infty
\end{equation}
Subresults
| ▶ | R5169: Infinite product of real numbers in right-closed unit interval does not vanish iff infinite sum of duals converges |