Proving "Bounds of log(n)" Inequality

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In summary, the conversation discusses the problem of proving an inequality involving an integer n and the logarithm function. The speaker suggests using a series expansion for the logarithm and mentions a pictoral representation that helped them understand the problem. They also mention attempting to use bounding step functions in their proof and receiving advice to look at the limiting difference between the harmonic series and natural logarithm. The speaker is still struggling with one side of the inequality and is unsure if their approach is correct. They ask for further hints or tips.
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Deano10
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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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  • #3
Deano10 said:
I can see pictorally why the inequality holds true
What is the pictoral reason?
 
  • #4
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.
 
  • #5
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...
 

FAQ: Proving "Bounds of log(n)" Inequality

What is the "Bounds of log(n)" inequality?

The "Bounds of log(n)" inequality is an expression used to describe the upper and lower limits of the logarithm function. It helps to determine the maximum and minimum values that can be obtained by the logarithm of a given number.

Why is proving the "Bounds of log(n)" inequality important?

Proving the "Bounds of log(n)" inequality is important because it helps to understand the behavior of the logarithm function and its relationship with other mathematical functions. It also has various applications in fields such as computer science, engineering, and economics.

What are the techniques used to prove the "Bounds of log(n)" inequality?

The most common techniques used to prove the "Bounds of log(n)" inequality are mathematical induction, direct proof, and contradiction. These techniques involve using logical reasoning and mathematical principles to establish the validity of the inequality.

Can the "Bounds of log(n)" inequality be generalized to other logarithmic functions?

Yes, the "Bounds of log(n)" inequality can be generalized to other logarithmic functions such as the natural logarithm and the base-10 logarithm. However, the specific bounds and proofs may vary depending on the base of the logarithm.

Are there any real-world applications of the "Bounds of log(n)" inequality?

Yes, the "Bounds of log(n)" inequality has various real-world applications, including analyzing the complexity of algorithms in computer science, predicting population growth in biology, and calculating interest rates and financial investments in economics.

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