ThmDex – An index of mathematical definitions, results, and conjectures.

Result R4695 on D993: Real number

An infinite product of real numbers on the open unit interval need not converge to zero

Formulation 0

Let $x_1, x_2, x_3, \ldots \in \mathbb{R}$ each be a D993: Real number such that

(i)	\begin{equation} x_n = e^{-2^{-n}} \end{equation}

Then

(1)	\begin{equation} x_1, x_2, x_3, \ldots \in (0, 1) \end{equation}
(2)	\begin{equation} \prod_{n = 1}^{\infty} x_n = e^{-1} \end{equation}

Formulation 1

Let $x_1, x_2, x_3, \ldots \in \mathbb{R}$ each be a D993: Real number such that

(i)	\begin{equation} x_n = e^{-1 / 2^n} \end{equation}

Then

(1)	\begin{equation} x_1, x_2, x_3, \ldots \in (0, 1) \end{equation}
(2)	\begin{equation} \lim_{N \to \infty} \prod_{n = 1}^N x_n = \frac{1}{e} \end{equation}

Also known as

Counterexample: sequence of real numbers on the open unit interval whose infinite product is nonzero

Proofs

Proof 0

Let $x_1, x_2, x_3, \ldots \in \mathbb{R}$ each be a D993: Real number such that

(i)	\begin{equation} x_n = e^{-2^{-n}} \end{equation}

Applying R3500: Homomorphism property of the standard natural real exponential function, we have \begin{equation} \begin{split} \prod_{n = 1}^N x_n = \prod_{n = 1}^N e^{-2^{-n}} = \exp \left( \sum_{n = 1}^N - 2^{-n} \right) = \exp \left( - \sum_{n = 1}^N \left( \frac{1}{2} \right)^n \right) \end{split} \end{equation} Result R3367: Characterisation of convergence of geometric basic real series shows that $\sum_{n = 1}^N ( 1 / 2 )^n \to 1$ as $N \to \infty$. Using result R5167: Standard natural real exponential function is sequentially continuous, we then have \begin{equation} \begin{split} \lim_{N \to \infty} \prod_{n = 1}^N x_n = \lim_{N \to \infty} \exp \left( - \sum_{n = 1}^N \left( \frac{1}{2} \right)^n \right) = \exp \left( - \lim_{N \to \infty} \sum_{n = 1}^N \left( \frac{1}{2} \right)^n \right) = \exp \left( - 1 \right) \end{split} \end{equation} $\square$