Testing for monotonicity

Alex Egg,

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 An , of my money each day, if we compare the values as a ratio we can determine if my money is increasing or decreasing:

an+1/an

Where an+1 is today and an is yesterday.

Here's an example w/ a real sequence:

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

so an= n/n+1 and an+1 = n+1/n+2 and simplify the ratio…

an+1/an = (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 an+1/an >1 and thus increasing.