Prove This Sequence Is Bounded Below by 0 Using Induction

In summary, the problem is that the inequality 1/(n+1) < log(n+1) - log(n) < 1/n cannot possibly be correct.
  • #1
shephard23
3
0

Homework Statement



Forgive my lack of LaTeX, not learned how to use it yet. Anyway, the problem is:

Use the inequalities

1/(n+1) < ln(n+1) - ln(n) < 1/n

to show that the sequence {xn} from n=1 to infinity defined by xn = 1 + 1/2 + ... + 1/n - ln(n) is strictly decreasing and bounded below by 0.

I've proved that it's strictly decreasing, stuck on the bounded below by 0 part.

Homework Equations


The Attempt at a Solution



We've been given a hint to use induction to show that xn > 0 for all natural numbers n so I'm going with that.

x(1) = 1 - ln(1) = 1 - 0 = 1 > 0. So it's true for n=1.

Assume it's true for n=k so x(k) = 1 + 1/2 + ... + 1/k - ln(k) > 0. I've tried rearranging this to 1 + 1/2 + ... + 1/k > ln(k) for use in the next step.

Now consider x(k+1) = 1 + 1/2 + ... + 1/k + 1/(k+1) - ln(k+1). I've to show this is greater than 0 if x(k) is greater than 0. The problem is that whenever I use the inequalities above I end up with x(k+1) is greater than something less than 0, for example:

1 + 1/2 + ... + 1/k + 1/(k+1) - ln(k+1)

> ln(k) + 1/(k+1) - ln(k+1) [using x(k)>0]

> 1(k+1) - 1/k [Using the inequality on the right above].

This is less than 0 so proves nothing, and most of my answers are coming out in this because I'm using similar methods. I thought I had to use the fact that x(k)>0 to show x(k+1)>0 so I've been trying it with no luck. Any advice at all?
 
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  • #2
This 1/(n+1) < log(n+1) - log(n) < 1/n inequality cannot possibly be correct .

Because from this you can deduce that
log(n+1) > log(n) + 1/(n+1)I tried log(100) - log (99) =0.004364

1/100 = .01

Already the inequality fails.
 
  • #3
My apologies, when we use log in our course it means the natural log. Sorry, I've been using it for so long it just became second nature. log in the above problem is the natural log, ln.
 
  • #4
I've managed to work out that x(k+1) > ln(k) - ln(k+1) + 1/k > 0 from the inequality on the right. Does anyone have any suggestions on getting

x(k+1) = 1 + 1/2 + ... + 1/k + 1/(k+1) > ln(k) - ln(k+1) + 1/k?

I'm having the same trouble as before.
 
  • #5
You will need to show

log(n)<1+1/2+1/3+...+1/n

Let's use the inequality log(n+1)<log(n)+1/n.

We obtain

log(n)<log(n-1)+1/(n-1)<log(n-2)+1/(n-2)+1/(n-1)<...

I think you'll get there this way...
 

FAQ: Prove This Sequence Is Bounded Below by 0 Using Induction

What is induction and how is it used in proving a sequence is bounded below by 0?

Induction is a mathematical proof technique that is used to prove a statement for all natural numbers. In the context of proving a sequence is bounded below by 0, induction is used to show that the statement holds for the first or base case, and then assuming it holds for some arbitrary index, the statement is shown to also hold for the next index.

Why is it important to prove that a sequence is bounded below by 0?

Proving that a sequence is bounded below by 0 is important because it ensures that the sequence will not drop below 0 and become negative. This is important in many applications, such as in physics or economics, where negative values may not make sense.

What are the steps involved in using induction to prove a sequence is bounded below by 0?

The steps involved in using induction to prove a sequence is bounded below by 0 are as follows:

  1. Show that the statement holds for the first or base case (usually n=0 or n=1).
  2. Assume that the statement holds for some arbitrary index k.
  3. Show that the statement also holds for the next index, k+1.
  4. Conclude that the statement holds for all natural numbers by the principle of mathematical induction.

Can induction be used to prove that a sequence is bounded below by a number other than 0?

Yes, induction can be used to prove that a sequence is bounded below by any number, not just 0. The steps involved would be the same, but instead of showing that the statement holds for the base case of 0, it would be shown for the base case of the specified number.

Are there any other methods besides induction that can be used to prove a sequence is bounded below by 0?

Yes, there are other methods that can be used to prove a sequence is bounded below by 0. These include the use of limits, the monotone convergence theorem, and the Cauchy criterion. However, induction is a commonly used and relatively simple method for proving this type of statement.

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