# Testing for monotonicity

*A sequence that is either increasing or decreasing is said to be monotone.*

This is a method to check if a given sequence is increasing or decreasing. I'm giving the intuition here for my own understanding.

### 1. Consider this analogy

Yesterday I had $100 and today I have $50. We can represent it as the ratio 50/100 or .5 which is less than 1

Yesterday I had $100 and today I have $100 still or 100/100 or 1.

Yesterday I had $100 and today I have $150 or 150/100 or 1.5 which is greater than 1.

Therefore we can say when my money is decreasing it's ratio is less than 1. When my money is constant it has a ratio of 1/1 and when my money is increasing it's ratio is > 1.</p>

### 2. Apply analogy to test sequences for monotonicity.

Given as sequence A_{n} , of my money each day, if we compare the values as a ratio we can determine if my money is increasing or decreasing:

`a`

_{n+1}/a_{n}

Where a_{n+1} is today and a_{n} is yesterday.

Here's an example w/ a real sequence:

`1/2, 2/3, 3/4, ..., n/n+1`

so `a`

and _{n}= n/n+1`a`

and simplify the ratio…_{n+1} = n+1/n+2

`a`

_{n+1}/a_{n} = (n+1)/(n+2) / n/(n+1) = n+1/n+2 * n+1/n = n^2+2n+1 / n^2+2n

From which we can see the numerator > denominator or `a`

and thus increasing._{n+1}/a_{n} >1

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testing-for-monotonicity

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math, calculus