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:
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
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.
Tags: math, calculus