Equivalence of Definitions of Absolute Convergence of Product
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Theorem
Let $\struct {\mathbb K, \norm {\,\cdot\,} }$ be a valued field.
Let $\sequence {a_n}$ be a sequence in $\mathbb K$.
The following definitions of the concept of Absolute Convergence of Product are equivalent:
Definition 1
The infinite product $\ds \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ is absolutely convergent if and only if $\ds \prod_{n \mathop = 1}^\infty \paren {1 + \norm {a_n} }$ is convergent.
Definition 2
The infinite product $\ds \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ is absolutely convergent if and only if the series $\ds \sum_{n \mathop = 1}^\infty a_n$ is absolutely convergent.
Proof
1 implies 2
By the Monotone Convergence Theorem, it suffices to show that the partial sums of $\ds \sum_{n \mathop = 1}^\infty a_n$ are bounded.
Because $\ds \prod_{n \mathop = 1}^\infty \paren {1 + \norm {a_n} }$ converges, its partial products are bounded.
By Bounds for Finite Product of Real Numbers:
- $\ds \sum_{n \mathop = 1}^N \norm {a_n} \le \prod_{n \mathop = 1}^N \paren {1 + \norm {a_n} }$
$\blacksquare$
2 implies 1
Proof 1
By the Monotone Convergence Theorem, it suffices to show that the partial products of $\ds \prod_{n \mathop = 1}^\infty \paren {1 + \norm {a_n} }$ are bounded.
By Bounds for Finite Product of Real Numbers:
- $\ds \prod_{n \mathop = 1}^N \paren {1 + \norm {a_n} } \le \map \exp {\sum_{n \mathop = 1}^N \norm {a_n} }$
Because $\ds \sum_{n \mathop = 1}^\infty \norm {a_n}$ converges, its partial sums are bounded.
$\blacksquare$
Proof 2
By the Monotone Convergence Theorem, it suffices to show that the partial products of $\ds \prod_{n \mathop = 1}^\infty \paren {1 + \norm {a_n} }$ are bounded.
By the AM-GM Inequality:
- $\ds \prod_{n \mathop = 1}^N \paren {1 + \norm {a_n} } \le \paren {\frac {N + \sum_{n \mathop = 1}^N \norm {a_n} } N }^N \le \paren {1 + \frac M N}^N$
where $M>0$ is such that $\ds \sum_{n \mathop = 1}^N \norm {a_n} \le M$ for all $N$.
By definition of the real exponential, $\paren {1 + \dfrac M N}^N \to \map \exp M$ as $N \to \infty$.
By Convergent Sequence in Metric Space is Bounded, $\paren {1 + \dfrac M N}^N$ is bounded.
$\blacksquare$