Bounds for Finite Product of Real Numbers
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Theorem
Let $a_1, a_2, \ldots, a_n$ be positive real numbers.
Then:
- $\ds \sum_{k \mathop = 1}^n a_k \le \prod_{k \mathop = 1}^n \paren {1 + a_k} \le \map \exp {\sum_{k \mathop = 1}^n a_k}$
Proof
Lower bound
Follows by expanding.
$\Box$
Upper Bound
Proof 1
By Exponential of x not less than 1+x:
- $\ds \prod_{k \mathop = 1}^n \paren {1 + a_k} \le \prod_{k \mathop = 1}^n \exp a_k = \map \exp {\sum_{k \mathop = 1}^n a_k}$
$\blacksquare$
Proof 2
By the AM-GM Inequality:
- $\ds \prod_{k \mathop = 1}^n \paren {1 + a_k} \le \paren {\frac {n + \sum_{k \mathop = 1}^n a_k} n}^n$
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$\blacksquare$