Summing sets of inverses of integers

In summary, the conversation discusses a set X of 16 distinct positive integers and the existence of an integer k and disjoint subsets A and B with specific properties. The goal is to find the smallest difference between two sums, \alpha and \beta, which involve fractions of the elements in A and B. The question is posed about which k, A, and B would be chosen for a specific set of 16 ordered integers, and the concept of variance is mentioned.
  • #1
Doom of Doom
86
0
How can I prove this?

Suppose [tex]X[/tex] is a set of 16 distinct positive integers, [tex]X=\left\{{x_{1}, \cdots , x_{16}}\right\}[/tex].
Then, for every [tex]X[/tex], there exists some integer [tex]k\in\left\{{1, \cdots , 8}\right\}[/tex] and disjoint subsets [tex]A,B\subset X[/tex]
[tex]A=\left\{a_{1},\cdots\ ,a_{k}\right\}[/tex] and [tex]B=\left\{b_{1},\cdots\ ,b_{k}\right\}[/tex]

such that [tex]\left|\alpha - \beta\right|<.00025[/tex],

where [tex]\alpha= \frac{1}{a_{1}}+\cdots+\frac{1}{a_{k}}[/tex] and [tex]\beta= \frac{1}{b_{1}}+\cdots+\frac{1}{b_{k}}[/tex].


I know that .00025 is pretty close to 2^-12.
 
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  • #2
Which k, A, B, would you choose for X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} ? It might be me, but I don't seem to find any.
 
  • #3
Well for a set of 16 ordered integers, how would you choose them to get the smallest difference? Think of variance of a set of data.
 

What is meant by "summing sets of inverses of integers"?

Summing sets of inverses of integers refers to the process of adding up the reciprocals of a group of whole numbers. The reciprocal of a number is simply 1 divided by that number.

Why is summing sets of inverses of integers important in science?

In science, summing sets of inverses of integers is often used in calculations involving series, sequences, and mathematical models. It can also be used in data analysis and statistical analysis.

What is the formula for summing sets of inverses of integers?

The formula for summing sets of inverses of integers is 1 + 1/2 + 1/3 + 1/4 + ... + 1/n, where n is the number of terms in the set.

What are some real-world applications of summing sets of inverses of integers?

Some real-world applications of summing sets of inverses of integers include calculating the average speed of an object, determining the resistance in an electrical circuit, and analyzing the growth rate of a population.

How does the sum of inverses of integers relate to the harmonic series?

The sum of inverses of integers is known as the harmonic series, which is a divergent series (meaning it does not have a finite sum). The value of the harmonic series increases as more terms are added, but it never reaches a finite value.

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