Math, asked by samiksha5055, 1 month ago

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Answered by user0888

131

$\large\underline{\text{Meaning of the expression}}$

Here is the explanation for each symbol.

$\displaystyle\red{\bullet}\ \sum^{n}_{k=1}\dfrac{1}{k}$ is the area of the rectangles.

$\displaystyle\red{\bullet}\ \ln n$ is the area bounded by $y=\dfrac{1}{x}$ , $x=1$ , $x=n$ and the x-axis.

$\displaystyle\red{\bullet}\ \lim_{x\to\infty}$ is the approaching value of an expression, as calculation repeats more and more times.

In the attachment, three areas are bounded by different colors.

$\displaystyle\red{\bullet}\ \text{Grey region: The area that the domain of }\ln n\text{ cannot cover.}$

$\displaystyle\red{\bullet}\ \text{Red region: The area under }y=\dfrac{1}{x}\text{.}$

$\displaystyle\red{\bullet}\ \text{Purple region: The difference between the curve and rectangle.}$

$\large\underline{\text{Note}}$

The number, $\displaystyle\gamma=\lim_{x\to\infty}(\sum^{n}_{k=1}\dfrac{1}{k}-\ln n)$ is Euler-Mascheroni constant.

$\large\underline{\text{Rational or irrational?}}$

It is not found whether the value is rational or irrational. By continued fraction method, it is proved that the denominator must be greater than $10^{244663}$ by Papanikolaou in 1997.

$\large\underline{\text{Graphical property}}$

The number has the following property about the area.

$\red{\bullet}\ \gamma=\text{(Area under the curve }\dfrac{1}{x})-\text{(Harmonic series)}$

This property is used in the approximation of the harmonic series, as $\gamma$ is around 0.577. And it is since the area of the purple region gets less and less. We get the following approximation.