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Version 9 |
| In the \PMlinkescapetext{complex analysis}, one often uses the definition of the convergence of infinite product $\prod_{k = 1}^{\infty}u_k$ where the case\, $\lim_{k\to\infty}u_1u_2\cdots u_k = 0$ is excluded.\, Then one has the |
\begin{thmplain} \,The infinite product $\prod_{k = 1}^{\infty}u_k$ of the non-zero complex numbers $u_1$, $u_2$, ... is convergent iff for every positive number $\varepsilon$ there exists a positive number $n_\varepsilon$ such that the condition |
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| \begin{thmplain}\, The infinite product $\prod_{k = 1}^{\infty}u_k$ of the non-zero complex numbers $u_1$, $u_2$, ... is convergent iff for every positive number $\varepsilon$ there exists a positive number $n_\varepsilon$ such that the condition |
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| $$|u_{n+1}u_{n+2}...u_{n+p}-1| < \varepsilon \quad \forall \,p\in\mathbb{Z}_+$$ |
$$|u_{n+1}u_{n+2}...u_{n+p}-1| < \varepsilon \quad \forall \,p\in\mathbb{Z}_+$$ |
| is true as soon as \,$n \geqq n_\varepsilon$. |
is true as soon as \,$n \geqq n_\varepsilon$. |
| \end{thmplain} |
\end{thmplain} |
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\textbf{Corollary.}\, If the infinite product converges, then we necessarily have\, $\lim_{k\to\infty}u_k = 1$.
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\textbf{Corollary.} \,If the infinite product converges, then we necessarily have \,$\lim_{k\to\infty}u_k = 1$.
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| When the infinite product converges, we say that the {\em value of the infinite product} is equal to $\lim_{k\to\infty} u_1u_2...u_k$. |
When the infinite product converges, we say that the {\em value of the infinite product} is equal to $\lim_{k\to\infty} u_1u_2...u_k$. |
