How to Prove 0.333... < x/y < 0.5 with $x$ and $y$ Defined as Fractions?

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In summary, we are given two fractions $y$ and $x$, with $y$ being a sum of fractions with denominators from 10 to 19 squared, and $x$ being a sum of fractions with denominators from 21 to 40 squared. We are asked to prove that $\dfrac{1}{3}\lt \dfrac{x}{y}\lt \dfrac{1}{2}$, which can be broken down into two parts: showing that $\dfrac{x}{y}\gt \dfrac{1}{3}$ and showing that $\dfrac{x}{y}\lt \dfrac{1}{2}$. The first part can be proven by showing that $3x\gt y$,
  • #1
anemone
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Let $y=\dfrac{1}{10^2}+\dfrac{1}{11^2}+\cdots+\dfrac{1}{19^2}$ and $x=\dfrac{1}{21^2}+\dfrac{1}{22^2}+\cdots+\dfrac{1}{40^2}$.

Prove that $\dfrac{1}{3}\lt \dfrac{x}{y}\lt \dfrac{1}{2}$.
 
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  • #2
anemone said:
Let $y=\dfrac{1}{10^2}+\dfrac{1}{11^2}+\cdots+\dfrac{1}{19^2}$ and $x=\dfrac{1}{21^2}+\dfrac{1}{22^2}+\cdots+\dfrac{1}{40^2}$.

Prove that $\dfrac{1}{3}\lt \dfrac{x}{y}\lt \dfrac{1}{2}$.

let me prove the 2nd part that is

$\dfrac{x}{y}\lt \dfrac{1}{2}$
or $2x < y$

we have
$\dfrac{1}{n^2}=\dfrac{4}{(2n)^2} > \dfrac{2}{(2n+1) ^2} +\dfrac{2}{(2n+2) ^2}$
by taking n from 10 to 19 and adding we get the result
 
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  • #3
anemone said:
Let $y=\dfrac{1}{10^2}+\dfrac{1}{11^2}+\cdots+\dfrac{1}{19^2}$ and $x=\dfrac{1}{21^2}+\dfrac{1}{22^2}+\cdots+\dfrac{1}{40^2}$.

Prove that $\dfrac{1}{3}\lt \dfrac{x}{y}\lt \dfrac{1}{2}$.
$\dfrac{20}{861}=\dfrac {1}{21}-\dfrac{1}{41}<x<\dfrac {1}{20}-\dfrac{1}{40}=\dfrac{1}{40}\\$
$\dfrac{40}{861}<2x<\dfrac {1}{20}\\$
$\therefore \dfrac {1}{2x}>20----(A)\\$
$3x>\dfrac{60}{861}---(B)\\$
$\dfrac{1}{20}=\dfrac {1}{10}-\dfrac{1}{20}<y<\dfrac {1}{9}-\dfrac{1}{19}=\dfrac{10}{171}\\$
$\therefore y>\dfrac{1}{20}----(C)\\$
$\dfrac {1}{y}>\dfrac{171}{10}---(D)$
$(C)\times(A)>1$, and $(B)\times (D)>1$
the proof of both sides is finished
 
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  • #4
Albert said:
$\dfrac{20}{861}=\dfrac {1}{21}-\dfrac{1}{41}<x<\dfrac {1}{20}-\dfrac{1}{40}=\dfrac{1}{40}\\$

How ?
 
  • #5
kaliprasad said:
How ?
$\sum (\dfrac {1}{n}-\dfrac {1}{n+1})=\sum \dfrac {1}{n(n+1)}<\sum \dfrac {1}{n^2}<\sum \dfrac {1}{(n-1)(n)}=\sum (\dfrac {1}{n-1}-\dfrac {1}{n})$
 
  • #6
Thanks Albert for participating and your method is correct.

Thanks to kaliprasad as well for proving half of the problem.:)
 
  • #7
My solution:

First notice that $22^2>21^2>4(10^2),\,24^2>23^2>4(11^2),\cdots,\,40^2>39^2>4(19^2)$ we get $\dfrac{1}{4(10^2)}+\dfrac{1}{4(10^2)}+\dfrac{1}{4(11^2)}+\dfrac{1}{4(11^2)}+\cdots+\dfrac{1}{4(19^2)}+\dfrac{1}{4(19^2)}>\dfrac{1}{21^2}+\dfrac{1}{22^2}+\cdots+\dfrac{1}{40^2}$, i.e. $\dfrac{1}{2}\left(y\right)>x$ or simply $y>2x$.

On the other hand, we have $6(10^2)>22^2>21^2,\,6(11^2)>24^2>23^2,\cdots,\,6(19^2)>40^2>39^2$ we get $\dfrac{1}{6(10^2)}+\dfrac{1}{6(10^2)}+\dfrac{1}{6(11^2)}+\dfrac{1}{6(11^2)}+\cdots+\dfrac{1}{6(19^2)}+\dfrac{1}{6(19^2)}<\dfrac{1}{21^2}+\dfrac{1}{22^2}+\cdots+\dfrac{1}{40^2}$, i.e. $\dfrac{1}{3}\left(y\right)<x$ or simply $y<3x$.

Combining both results we get $\dfrac{1}{3}\lt \dfrac{x}{y}\lt \dfrac{1}{2}$ and we're hence done.
 

FAQ: How to Prove 0.333... < x/y < 0.5 with $x$ and $y$ Defined as Fractions?

What does "Prove 0.333...

This statement means that there is a decimal number between 0.333... (repeating) and 0.5, and we want to prove that it is possible to write this number as a fraction with x as the numerator and y as the denominator.

What is the significance of proving this statement?

Proving this statement is important because it shows that there is a rational number (a number that can be written as a fraction) between 0.333... and 0.5. It also demonstrates the concept of infinite decimals and how they can be represented as fractions.

How can this statement be proven?

This statement can be proven using mathematical techniques such as the Archimedean property, which states that given any two real numbers, there is always a rational number between them. Additionally, we can use algebraic manipulation to find a specific fraction that satisfies the given inequality.

Is there a specific value for x and y that will satisfy this statement?

No, there are infinite possible values for x and y that will satisfy this statement. However, any value of x and y that satisfies the inequality and produces a decimal between 0.333... and 0.5 can be considered a valid proof.

Can this statement be disproven?

No, this statement cannot be disproven. The existence of rational numbers between two given numbers has been mathematically proven, and the statement itself is based on this fact. Therefore, it cannot be disproven.

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