Can You Prove This Inequality Involving Fractions and Square Roots?

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    2015
In summary, to prove an inequality involving fractions and square roots, you can simplify both sides of the inequality using algebraic operations and then use properties of inequalities and mathematical techniques such as induction or contradiction to formally prove it. While a calculator can be used to check work, it is not a substitute for a formal proof. Common mistakes to avoid include assuming the inequality holds for all values and making errors in simplification. The use of both fractions and square roots is not always necessary, and other mathematical concepts like the triangle inequality can also be used in the proof.
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anemone
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Prove that $1-\dfrac{1}{2014}\left(\dfrac{1}{2}+\dfrac{1}{3}+\cdots+\dfrac{1}{2015}\right)>\dfrac{1}{\sqrt[2014]{2015}}$


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No one answered last week's problem.:(

You can view the proposed solution below:

Note that we could rewrite the LHS of the given inequality as

$\begin{align*}1-\dfrac{1}{2014}\left(\dfrac{1}{2}+\dfrac{1}{3}+\cdots+\dfrac{1}{2015}\right)&=\dfrac{1}{2014}\left(\overbrace{1+1+1+\cdots+1}^{2014}-\left(\dfrac{1}{2}+\dfrac{1}{3}+\cdots+\dfrac{1}{2015}\right)\right)\\&=\dfrac{1}{2014}\left(\left(1-\dfrac{1}{2}\right)+\left(1-\dfrac{1}{3}\right)+\left(1-\dfrac{1}{4}\right)\cdots+\left(1-\dfrac{1}{2014}\right)+\left(1-\dfrac{1}{2015}\right)\right)\\&=\dfrac{1}{2014}\left(\dfrac{1}{2}+\dfrac{2}{3}+\dfrac{3}{4}+\cdots+\dfrac{2013}{2014}+\dfrac{2014}{2015}\right)\end{align*}$

At this point, we can apply the AM-GM inequality to

$\begin{align*}\dfrac{1}{2}+\dfrac{2}{3}+\dfrac{3}{4}+\cdots+\dfrac{2013}{2014}+\dfrac{2014}{2015}&\ge 2014\left(\dfrac{1}{2}\cdot\dfrac{2}{3}\cdot\dfrac{3}{4}\cdots\dfrac{2013}{2014}\cdot\dfrac{2014}{2015}\right)^{\frac{1}{2014}}\\& \ge 2014\left(\frac{1}{2015}\right)^{\frac{1}{2014}}\end{align*}$

Therefore

$\begin{align*}1-\dfrac{1}{2014}\left(\dfrac{1}{2}+\dfrac{1}{3}+\cdots+\dfrac{1}{2015}\right)&=\dfrac{1}{2014}\left(\dfrac{1}{2}+\dfrac{2}{3}+\dfrac{3}{4}+\cdots+\dfrac{2013}{2014}+\dfrac{2014}{2015}\right)\\&\ge \dfrac{1}{2014}\left(2014\left(\frac{1}{2015}\right)^{\frac{1}{2014}}\right)\\&\ge \left(\frac{1}{2015}\right)^{\frac{1}{2014}}\text{Q.E.D.}\end{align*}$
 

FAQ: Can You Prove This Inequality Involving Fractions and Square Roots?

How do you prove an inequality involving fractions and square roots?

To prove an inequality involving fractions and square roots, you can start by simplifying both sides of the inequality using basic algebraic operations. Then, you can use the properties of inequalities such as the addition, subtraction, multiplication, and division properties to manipulate the inequality into a form that is easier to prove. Finally, you can use mathematical techniques such as induction, contradiction, or direct proof to formally prove the inequality.

Can I use a calculator to prove the inequality?

While a calculator can be a helpful tool to check your work, it is not a substitute for a formal mathematical proof. Proving an inequality involving fractions and square roots requires logical reasoning and mathematical techniques, which cannot be simply verified by a calculator.

What are the common mistakes to avoid when proving an inequality involving fractions and square roots?

One common mistake is to assume that the inequality holds true for all values of the variables without considering any restrictions or special cases. It is also important to be careful when simplifying expressions involving fractions and square roots, as mistakes in simplification can lead to incorrect proofs. Additionally, do not forget to justify each step of your proof using mathematical properties and techniques.

Is it necessary to use both fractions and square roots in the inequality?

No, it is not necessary to use both fractions and square roots in the inequality. You may encounter inequalities that involve only fractions or only square roots. The approach to proving these types of inequalities may differ, but the same principles and techniques apply.

Can I use other mathematical concepts to prove the inequality?

Yes, you can use other mathematical concepts such as the triangle inequality, Cauchy-Schwarz inequality, or AM-GM inequality to prove an inequality involving fractions and square roots. However, be sure to clearly state and justify the use of these concepts in your proof.

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