Bounds for Finite Product of Real Numbers

Theorem
Let $a_1,\ldots,a_n$ be positive real numbers.

Then:
 * $\displaystyle \sum_{k \mathop = 1}^n a_k \leq \prod_{k \mathop = 1}^n\left({1 + a_k}\right) \leq \exp\left( \sum_{k \mathop = 1}^n a_k\right)$

Lower bound
Follows by expanding.

Proof 1
By Exponential of x not less than 1+x:
 * $\displaystyle \prod_{k \mathop = 1}^n\left({1 + a_k}\right) \leq \prod_{k \mathop = 1}^n\exp a_k = \exp\left( \sum_{k \mathop = 1}^na_k\right)$

Proof 2
By the AM-GM Inequality:
 * $\displaystyle \prod_{k \mathop = 1}^n\left({1 + a_k}\right) \leq \left(\frac{n + \sum_{k \mathop = 1}^na_k}n \right)^n$

Also see

 * Weierstrass Product Inequality