Proving an+1 < an: A Derivative or Logical Approach?

[ronaldor9]

Suppose I want to show that a_n+1 < a_n one could find the derivative of the sequence and then show that it is negative for a certain domain of values for x. However is it also possible to conclusively prove that a_n+1 < a_n simply by subbing in n+1 and n and show that the inequality leads to a logical conclusion?

 

[n!kofeyn]

Do you have a specific example that you are wanting help with maybe? In general, it is just like showing any other inequality. For example, let a_n=-n. Then since -(n+1)<-n, we know that a_n+1<a_n.

 

[ronaldor9]

Sure, how about this: Determine if a_n+1 is < or = to a_n
$$a_n=\frac{n}{2^{n-1}}$$

My concern is if it is possible to assume:
$$\frac{n+1}{2^n} \leq \frac{n}{2^{n-1}}$$, and show that it leads to the correct statement,
$$\frac{1}{2} \leq \frac{n}{n+1}, \qquad n\geq 1$$?

 

[HallsofIvy]

Suppose I want to show that a_n+1 < a_n one could find the derivative of the sequence and then show that it is negative for a certain domain of values for x. However is it also possible to conclusively prove that a_n+1 < a_n simply by subbing in n+1 and n and show that the inequality leads to a logical conclusion?

No, it is NOT possible to prove anything by assuming what you want to prove and showing that you can prove something "logical" from it. You can prove ANYTHING from a FALSE hypothesis.

What you can do, of course, is start from the definition of your sequence and derive "a_{n+1[/sup]< an".}

 

[Office_Shredder]

Sure, how about this: Determine if a_n+1 is < or = to a_n
$$a_n=\frac{n}{2^{n-1}}$$

My concern is if it is possible to assume:
$$\frac{n+1}{2^n} \leq \frac{n}{2^{n-1}}$$, and show that it leads to the correct statement,
$$\frac{1}{2} \leq \frac{n}{n+1}, \qquad n\geq 1$$?

In fact, you did your proof backwards. A correct proof would be starting with

$$\frac{1}{2} \leq \frac{n}{n+1}, \qquad n\geq 1$$

which you know is correct

and concluding


$$\frac{n+1}{2^n} \leq \frac{n}{2^{n-1}}$$

Which you weren't sure (until now) whether it is correct

 

[ronaldor9]

It seems kinda like cheating because you would have to still find out that $$\frac{1}{2} \leq \frac{n}{n+1}, \qquad n\geq 1$$ by first guessing and playing around?

I mean why do you have to start off backwards it seems like in almost all cases where i have used this method (of guessing the inequality and then fixing it to yield a correct statement) all the time the steps are reversible and lead to correct conclusion both ways, i cannot see a time where this would not be correct. (maybe by some division by zero, but i have not encountered such a case)

 

[JG89]

It seems kinda like cheating because you would have to still find out that $$\frac{1}{2} \leq \frac{n}{n+1}, \qquad n\geq 1$$ by first guessing and playing around?

I mean why do you have to start off backwards it seems like in almost all cases where i have used this method (of guessing the inequality and then fixing it to yield a correct statement) all the time the steps are reversible and lead to correct conclusion both ways, i cannot see a time where this would not be correct. (maybe by some division by zero, but i have not encountered such a case)

You wouldn't guess and play around to derive the inequality $$\frac{1}{2} \le \frac{n}{n+1}$$ for n >= 1.

If n >= 1,then 1/n <= 1. Add 1 to both sides then multiply both sides by n, which gives n + 1 <= 2n, implying that 1/2 <= n/(n+1).

As for your other question, consider this example. 1 < 2. Multiplying by 0 gives 0 < 0, which is obviously not true. If you are going to assume the premise, then reverse your steps to arrive at a valid proof, you have to be VERY careful about what you do. In fact once you finish reversing the steps and writing down your proof, you should look over the proof to make sure you haven`t done anything silly, like what I did in my counter-example.

 

[Edgardo]

I mean why do you have to start off backwards it seems like in almost all cases where i have used this method (of guessing the inequality and then fixing it to yield a correct statement) all the time the steps are reversible and lead to correct conclusion both ways, i cannot see a time where this would not be correct. (maybe by some division by zero, but i have not encountered such a case)

You have to be careful. I've once made the error myself.
Suppose you have to prove some statement S. Then it is not a valid proof to:
(1) start with S
(2) do some manipulations
(3) and arrive at a true statement.

For example:
(1) Statement S: 5 = 4
(2) Manipulations: 0*5 = 4*0
=> 0 = 0
(3) Since we arrived at a true statement (0 = 0 is true), the statement S is correct.

But the above is of course wrong! This is what HallsOfIvy mentioned.

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The way you start proofs is to write down FIRST what you already know is true, e.g.
we first write down n>=1 (as JG89 mentioned).

It seems a little "magic" and you could ask why I should start with n>=1. Of course, you figured this out by "working backwards" from 1/2 <= n/(n+1). But still, you write down a true statement first and derive something from it.

In mathematics you will often see proofs where they start with some "magical" assumption. For example in delta-epsilon proofs they just start with some delta and you ask yourself how they thought of that delta. Of course, they worked backwards, but you don't include this "working backwards" in your proof.

 

[Дьявол]

First try to find the sign of $a_n-a_{n+1}$ i.e positive or negative.

$$a_n=\frac{n}{2^{n-1}}$$

$$a_n-a_{n+1}=\frac{n}{2^{n-1}}-\frac{n+1}{2^{n}}=\frac{2n}{2^n}-\frac{n}{2^n}-\frac{1}{2^n}=\frac{n-1}{2^n}$$

$$\frac{1}{2^n} \geq 0$$

and

since $n\geq 1$, also $n-1 \geq 0$

so both are positive, and $+/+=+$

and $a_n-a_{n+1} > 0$ so $a_n>a_{n+1}$

Regards.

 

[ronaldor9]

It now makes more sense to me, at the start I thought it is useless beacuse of course you would have had to reach the conclusion $$\frac{1}{2} \leq \frac{n}{n+1}, \qquad n\geq 1$$ by working forwards (most likely, at least this way is the simplest), and then again work backwards to actually prove it true. I guess this is what first confused me beacuse it seemed almost conclusive that every step is reversible, and therefore I saw starting from the back then working a waste of time, but now i know you cannot take this assumption for granted as Edgardo points out in his example.