# Proving Series Inequality: $\sqrt[3]{\frac{2}{1}}$ to $\frac{1}{8961}$

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• anemone
In summary, the purpose of proving series inequality is to demonstrate a pattern or formula in a mathematical series and understand its behavior and relationship to other concepts. It can be proven using various techniques such as mathematical induction, comparison test, and limit comparison test. The series in the inequality represents a sequence of numbers being compared. Series inequality can be applied to other mathematical concepts and has real-world applications in fields such as engineering, physics, and economics. It can be used to analyze and predict the behavior of complex systems and solve optimization problems.
anemone
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MHB
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Prove that $\sqrt[3]{\dfrac{2}{1}}+\sqrt[3]{\dfrac{3}{2}}+\cdots+\sqrt[3]{\dfrac{996}{995}}-\dfrac{1989}{2}<\dfrac{1}{3}+\dfrac{1}{6}+\cdots+\dfrac{1}{8961}$.

Note that

$\dfrac{3k+1}{3k}+\dfrac{3k+2}{3k+1}+\dfrac{3k+3}{3k+2}=3+\dfrac{1}{3k}+\dfrac{1}{3k+1}+\dfrac{1}{3k+2}$ and

$\dfrac{3k+1}{3k}\cdot\dfrac{3k+2}{3k+1}\cdot\dfrac{3k+3}{3k+2}=\dfrac{k+1}{k}$,

by the Arithmetic-Geometric Mean inequality, we have

$3+\dfrac{1}{3k}+\dfrac{1}{3k+1}+\dfrac{1}{3k+2}>3\sqrt[3]{\dfrac{k+1}{k}}$

Then

$\displaystyle 3\sum_{k=1}^{995} \sqrt[3]{\dfrac{k+1}{k}}<3\sum_{k=1}^{995}\left( 3+\dfrac{1}{3k}+\dfrac{1}{3k+1}+\dfrac{1}{3k+2}\right)$

and hence

\displaystyle\begin{align*} 3\sum_{k=1}^{995} \sqrt[3]{\dfrac{k+1}{k}}-\dfrac{1989}{2}&<995-\dfrac{1989}{2}+\sum_{k=1}^{995}\dfrac{1}{3}\left( \dfrac{1}{3k}+\dfrac{1}{3k+1}+\dfrac{1}{3k+2}\right)\\&=\dfrac{1}{2}+\left(\dfrac{1}{9}+\dfrac{1}{12}+\cdots+\dfrac{1}{8961}\right)\\&=\dfrac{1}{3}+\dfrac{1}{6}+\left(\dfrac{1}{9}+\dfrac{1}{12}+\cdots+\dfrac{1}{8961}\right)\end{align*}

## 1. What is the purpose of proving series inequality?

The purpose of proving series inequality is to establish a relationship between two series or sequences, and to determine which series is greater or smaller. This can help in solving problems related to calculus, statistics, and other fields of science.

## 2. How do you prove series inequality?

To prove series inequality, you need to show that the terms of one series are always greater or smaller than the terms of the other series. This can be done by using mathematical techniques such as induction, comparison tests, and limit comparison tests.

## 3. What is the significance of the series inequality $\sqrt[3]{\frac{2}{1}}$ to $\frac{1}{8961}$?

The series inequality $\sqrt[3]{\frac{2}{1}}$ to $\frac{1}{8961}$ is significant because it involves a radical expression and a fraction, which are commonly used in mathematical equations. By proving this inequality, we can gain a better understanding of how these types of expressions behave in comparison to each other.

## 4. Can series inequality be applied to real-life situations?

Yes, series inequality can be applied to real-life situations. For example, it can be used in economics to compare the growth rates of different investments, or in physics to compare the speeds of two moving objects. It is a useful tool in making quantitative comparisons in various fields.

## 5. Are there any limitations to proving series inequality?

Yes, there are limitations to proving series inequality. In some cases, it may be difficult to determine the exact relationship between two series, or the series may have complex terms that make it challenging to compare. Additionally, the proofs may require advanced mathematical knowledge and techniques, making it inaccessible to those without a strong mathematical background.

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