Prove the inequality - 1\4(ln2)^2 <= sigma(2^n\(2^(2^n)))

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In summary, the conversation discusses a question from an old exam that requires proving an inequality involving a series. The person initially solved it by calculating the first three terms of the series, but is unsure if it is the proper way to solve it. They double checked their work and confirmed it is correct, but still think there may be another direction. Another person comments on how surprising it is that the left side is not close to the right side, even with limited accuracy in the calculation. They suggest looking for a non-computational proof.
  • #1
puzzek
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Homework Statement


This is a question taken from an old exam so I am not sure to which subject in calculus it's connected to...

Prove the inequality:
[itex]\frac{1}{4(ln2)^2}[/itex][itex]\leq[/itex][itex]\sum\frac{2^n}{2^(2^n)}[/itex]

(sigma is from 1 to +inf, and the Denominator on the right side is (2^(2^n))

Homework Equations





The Attempt at a Solution



Well, I solved it by calculating the first 3 elements of the sum showing their sum is larger than the left side, and that proves it. But, I don't think that's the way I suppose to solve it!
(I thought it somehow connected to derivative of the sum of the integrals of the function in the sigma.

Any help would be greatly appreciated.
 
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  • #2
There is nothing wrong with what you've done (noting that all terms of the series are positive). However, if it is that trivial, I would double check that you have the left hand side right. I don't see that a teacher would assign something that trivial.
 
  • #3
PAllen,

Thank you for your response.

I double checked it again and it's exactly as it is written on my paper.

By the way, this is not that trivial, because in our test you are not allowed to use a calculator hence it's hard to calculate the value of (ln2)^2. One can still use some knowledge to evaluate it, but during an exam it's not that easy.

I still think I got it wrong and that's not the "proper" way to solve it. Is there any other direction?
 
  • #4
puzzek said:
PAllen,

Thank you for your response.

I double checked it again and it's exactly as it is written on my paper.

By the way, this is not that trivial, because in our test you are not allowed to use a calculator hence it's hard to calculate the value of (ln2)^2. One can still use some knowledge to evaluate it, but during an exam it's not that easy.

I still think I got it wrong and that's not the "proper" way to solve it. Is there any other direction?

What is strange is how 'not close' the left side is. If you only use ln2 to 3 significant digits, and do computations guaranteed to be bounded above on the left and below on the right, you still trivially prove the inequality with 3 terms. I'll have to leave it so someone else to propose a non-computational proof. I don't see one at the moment.
 

FAQ: Prove the inequality - 1\4(ln2)^2 <= sigma(2^n\(2^(2^n)))

1. What is the purpose of proving this inequality?

The purpose of proving this inequality is to demonstrate a mathematical relationship between two quantities, in this case the expression 1/4(ln2)^2 and the summation of a series involving powers of 2. This can help to understand the magnitude of the difference between the two quantities and potentially provide insights into other mathematical problems.

2. How do you begin to prove this inequality?

To prove this inequality, one would typically start by simplifying the expression on both sides and looking for any patterns or relationships. Then, mathematical techniques such as induction, substitution, or manipulation of inequalities can be used to show that the two quantities are indeed related in the desired manner.

3. What is the significance of the 1/4 term in the left side of the inequality?

The 1/4 term on the left side of the inequality serves as a scaling factor, adjusting the magnitude of the logarithmic term to be more comparable to the summation on the right side. This allows for a more direct comparison and can help to prove the inequality.

4. Are there any specific values of n for which this inequality holds true?

Yes, this inequality holds true for all positive values of n. However, the specific values of n may affect the magnitude of the difference between the two quantities, and thus may require different approaches to proving the inequality.

5. Can this inequality be used in other areas of mathematics or science?

Yes, this inequality may have applications in various fields such as number theory, probability, and computer science. It may also serve as a useful tool for solving other mathematical problems that involve powers of 2 or logarithmic functions.

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