Proving "Bounds of log(n)" Inequality

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Homework Help Overview

The problem involves proving an inequality related to the logarithm function and the harmonic series, specifically that for any integer n >= 2, the sum of the series from 1/2 to 1/n is bounded by log(n), and that log(n) is also bounded above by the sum from 1 to 1/(n-1).

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants discuss visualizing the inequality through graphical representations, particularly using step functions to illustrate bounds. There is mention of adapting existing proofs related to the harmonic series and logarithms, but uncertainty remains about the appropriateness of this approach.

Discussion Status

The discussion is ongoing, with participants sharing insights and seeking clarification on their reasoning. Some guidance has been offered regarding potential methods, but no consensus has been reached on a definitive approach to the proof.

Contextual Notes

Participants express uncertainty about the validity of their approaches and the need for hints or tips to progress further in the proof. There is an acknowledgment of the complexity of proving both sides of the inequality.

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Homework Statement



Prove that for any integer n >= 2,

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


Homework Equations



None


The Attempt at a Solution



I can see pictorally why the inequality holds true but despite numerous am struggling to make any real progress! Any hints or tips on how to get started would be very much appreciated!
 
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Deano10 said:
I can see pictorally why the inequality holds true
What is the pictoral reason?
 
The picture I had in mind was of that of log (n) with step functions (of the values in the inequalities) both above and below the graph drawn out by log (n).

It is this that leads me to think the proof must involve the use of bounding step functions, but I cannot see how to begin.
 
I have had a further look at this and think that I can adapt the proof of the fact that the limiting difference between the harmonic series and natural logarithm tending to the Euler constant to prove one side of the inequality.

However, this still leaves the other side of the inequality unsolved and the fact that I am not sure this is the approach I should be taking!

Any hints would be most appreciated...
 

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