Proving Monotonic Sequence: Diff & Examples

[jokerzz]

I have 2 questions. How do you use differetiation to prove whether sequence is monotonic? For example: 1/n+ln(n)

My 2nd question is, how do you prove whether sequence is EVENTUALLY monotonic?

 

[ystael]

1. How do you use differentiation to determine whether a function defined on a subset of $$\mathbb{R}$$ is monotonic? Can you find a function whose values at the natural numbers give the terms of your sequence?

2. "Eventually monotonic" just means "monotonic after some large index $$N$$". So take any condition on the index $$n$$ that you would use to prove a sequence is monotonic, and verify the weaker statement that you can find some $$N > 0$$ such that the condition holds whenever $$n > N$$.

 

[jokerzz]

your answer makes no sense to me! I don't think you have to be soooo damn cryptic

 

[mistermath]

What does it mean for a function to be monotonic? It's either constant (everywhere or at some intervals), and at the intervals that it is not constant it is either:
1. increasing or
2. decreasing

but this is an exclusive or, meaning if at some interval it is increasing, it will never be decreasing.

In math terms:
if x $$\leq$$ y then f(x) $$\leq$$ f(y) OR
if x $$\geq$$ y then f(x) $$\geq$$ f(y)

So what you need to do for 1 is: assume x $$\leq$$ y then look at f(x) and compare that to f(y). See what the derivatives look like etc.. depending on your level, it may be something as simple as since the derivative is always positive, this function is always increasing etc (or in this case decreasing.. I don't know, look at the graph to figure that out).

For 2 (again assuming your math level is not super high and that's why you didn't understand ystael), find a point on the real number line where the function is monotonic after that point, then use that to help you. work on #1 first, then that'll help you with #2

Also don't be RUDE!

 

[jokerzz]

So ur saying for question 2 its trial and error? Isnt there any method like a(n+1)-an<0 or sumthing?

 

[mistermath]

For question 2 you use trial and error to find your N (the point at which the function becomes monotonic).

You cannot prove that a general function is eventually monotonic.. because not every function is. (Take sin(x) for example, it fluctuates forever!)