Albert said:prove :
$\dfrac {1\times 3 \times 5\times------\times 99}{2\times 4\times 6\times --\times {100}}>\dfrac {1}{10\sqrt 2}$
very nice !kaliprasad said:Let
$A= \frac{3}{4}*\frac{5}{6}\cdots\frac{99}{100}$
$B = \frac{2}{3}*\frac{4}{5}\cdots\frac{98}{99}$
In the above products each term of A is > B pairwise so A > B
$A * B = \frac{2}{100}$ and hence $A \gt\sqrt{(\frac{2}{100})}$
hence $ A \gt\frac{\sqrt{2}}{10}$
hence given expression which is $\frac{A}{ 2}$
so $\gt\dfrac{1}{10\sqrt{2}}$
Proof: Inequality of Series is a mathematical concept that states that if two series have the same terms, but in a different order, and one of the series converges, then the other series also converges. This means that the order of terms in a convergent series does not affect its convergence.
Understanding Proof: Inequality of Series is important because it allows us to manipulate and rearrange series in order to simplify and solve mathematical problems. It also helps us to determine the convergence or divergence of a series, which is crucial in many mathematical and scientific applications.
The difference between Proof: Inequality of Series and Proof: Convergence of Series is that Inequality of Series only applies when the terms of two series are exactly the same, but in a different order. On the other hand, Convergence of Series applies to any series, regardless of the order of terms.
Proof: Inequality of Series has many real-life applications, such as in finance, where it is used to calculate interest rates and compound interest. It is also used in physics and engineering to solve problems related to series and sequences, such as in the study of circuits and vibrations.
If your research or experiments involve series or sequences, Proof: Inequality of Series can be used to analyze and manipulate data, as well as to determine the convergence or divergence of a series. It can also be used to simplify and solve mathematical equations that involve series or sequences.