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Questions tagged [euler-mascheroni-constant]

For questions related to Euler's constant $\gamma$, which is defined to be the limiting difference between the natural logarithm and the harmonic series.

Euler's constant, also called the Euler-Mascheroni constant and typically denoted $\gamma$, is defined to be the limiting difference between the natural logarithm and the harmonic numbers:

$$\gamma=\lim_{n \to \infty}H_n-\log n$$ where

$$H_n=1+\frac{1}{2}+\cdots+\frac{1}{n}$$

Euler's constant arises in analysis and number theory, in part due to its connections with the gamma and zeta functions.

Note this is not the same as Euler's number $e$, defined by $e:=\sum_{n=0}^{\infty}\frac{1}{n!}$. Questions about this number should use the tag .

Source: the Wikipedia article on the Euler-Mascheroni constant.

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Integral representation of the Euler-Mascheroni constant involving $\pi$

A month ago, I came up with a proof that $\gamma = \frac12 + \int_0^{\frac1\pi} \arctan(\cot(\frac1x)) \,dx$ where $\gamma$ is the Euler-Mascheroni constant and $\arctan$ is the inverse $\tan$ function. My proof is based on the idea, that $\lfloor…
Kinheadpump
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How to show $\gamma =\sum_{m=2}^{\infty}(-1)^{m}\frac{\zeta (m)}{m}$?

How to show $$\gamma =\sum_{m=2}^{\infty}(-1)^{m}\frac{\zeta (m)}{m}$$ where $\gamma $ is the Euler-Mascheroni constant and $\zeta (m)$ is the Riemann Zeta Function.
esege
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Why is this proof of the irrationality of $\gamma$ wrong?

By definition, $$\gamma=\lim_{n\to \infty}\left(\sum_{k=1}^n \left(\frac{1}{k}\right)-ln(n)\right)$$ It is clear that $\sum_{k=1}^n \left(\frac{1}{k}\right)$ will always be rational, as it is a sum of rational numbers. It can be showed that…
Juan Moreno
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How did Euler calculate his $\gamma$ constant to 16 digits?

I couldn't find any information online on how Euler calculated this to 16 digits of accuracy: $$ \gamma = - \ln{n} + \sum_{k=1}^\infty \frac1k $$ Obviously not like that, can somebody help please?
Betydlig
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Prove limit is the Euler-Mascheroni constant $\gamma$

How do I prove this result? $$\lim_{\varepsilon\to0^+}\left[\operatorname{li}(1-\varepsilon)-(1-\varepsilon)\ln\ln\frac{1}{1-\varepsilon}\right]=\gamma$$with$$\operatorname{li}(x):=\int_0^x\frac{dt}{\ln t}$$ Wolfram Alpha obtains this limit.
Charlie Cat
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How to show that Euler constant lies in $(0,1)$

Let $$ C_n=\sum_{i=1}^n \frac{1}{i}-\log n,\quad n\geq 1 $$ It is well-known that the limit of $(C_n)_{n\geq 1}$ exists, where its limit is denoted $\gamma$, Euler's constant. I'll shortly write what the author Artin has stated this proof in his…
Mr.MathDoctor
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Prove the de la Vallée-Poussin's formula

The Euler-Mascheroni constant is defined as $\gamma := \lim_{x\to\infty}(H_n - \ln\,n)$ where $H_n = \sum_{k=1}^{n}\frac{1}{k}$, and the de la Vallée-Poussin's formula states that: $$\gamma =…
user834302
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Table of integrals with Euler's constant gamma.

I'm working on a large table of integrals with Euler's constant $\gamma=0.577...$ It's a very interesting point, that a lot of integrals are found in similar tables like Gradshteyn-Ryzhik, but are not able to calculate with maple or mathematica. For…
skraemer
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Integral of the difference between a function and its floor

I know the following identity: $$\int_1^\infty\left(-\frac{1}{x}+\frac{1}{\lfloor x\rfloor}\right)dx=\gamma$$ I have a function $f(b)=0.5\left(\sqrt{(N-b^2)}-b+1\right)$. Can a similar identity be derived for $$\int_1^\infty\left(-f(b)+\lfloor…
RTn
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Help me understand e (exponential/euler's number) and how taking e to a certain power changes time period and return??

Ok, so I understand $e$ is just $\lim\limits_{n\to \infty}(1 + \frac{1}{n})^n$, and ends up being $2.71828\ldots$ It assumes the annual return you are getting is $100\%$ and is continuously compounded. I don't understand how if you take $e$ to some…
user6472523
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Trying to solve a demonstration with the constant of Euler

I think this will be an easy problem for you, but I do not see the solution. I know that $$\displaystyle\sum_{k=0}^{\infty} \frac{1}{k!}=e$$ Knowing this, how can I demonstrate this $$\displaystyle\sum_{k=1}^{\infty}…
Muradin Bronzebeard
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Why did the natural constant e had to be 2.718..

I tried hard to not make this duplicate but if I have missed it please flag this question. So people constantly ask and answer why e is very important, where it came from, and its derivations, etc. "Oh it's so special because it appears here and…
Seung
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Flaw in this reasoning

$\ e^{i 2\pi} = 1,$ $\ e^{0} = 1$ $\Rightarrow $ $\ e^{i 2\pi} = e^{0}$ $\Rightarrow $ $\ {i 2\pi} = 0 $ $\Rightarrow $ $\ i = 0 $
bhaskarc
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What is the simplification of $ {e^2}^n $?

I'm stuck on this problem, can someone tell me what the simplification of the following is? $$\ {e^2}^n $$
random_x_y_z
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What happens if I divide $x$ by $\infty$?

The definition of $e$ is if $n$ is equal to $\infty$ By that definition can I algebraic manipulate the equation and say that $r/\infty=e^{(1/\infty)}-1$?
coderhk
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