# Bounds for Finite Product of Real Numbers

Jump to navigation
Jump to search

This page has been identified as a candidate for refactoring of basic complexity.In particular: Separate subpages for upper and lower bound for consistent referencingUntil this has been finished, please leave
`{{Refactor}}` in the code.
Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Refactor}}` from the code. |

## 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:

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

A specific link is needed here.In particular: link to an article that this is less than $\map \exp {\sum a_k}$, i.e. that this expression is increasing with $n$You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by searching for it, and adding it here.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{LinkWanted}}` from the code. |

$\blacksquare$