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:

$\displaystyle \sum_{k \mathop = 1}^n a_k \le \prod_{k \mathop = 1}^n \paren {1 + a_k} \le \map \exp {\sum_{k \mathop = 1}^n a_k}$


Proof

Lower bound

Follows by expanding.

$\Box$


Upper Bound

Proof 1

By Exponential of x not less than 1+x:

$\displaystyle \prod_{k \mathop = 1}^n \paren {1 + a_k} \le \prod_{k \mathop = 1}^n \exp a_k = \map \exp {\sum_{k \mathop = 1}^n a_k}$

$\blacksquare$


Proof 2

By the AM-GM Inequality:

$\displaystyle \prod_{k \mathop = 1}^n \paren {1 + a_k} \le \paren {\frac {n + \sum_{k \mathop = 1}^n a_k} n}^n$

$\blacksquare$


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