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shephard23
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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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