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.
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How to solve this post? [closed]
How long does it take to download a two-hour HD movie
from the iTunes store? According to Apple’s technical support
site, support.apple.com/en-us/HT201587, downloading such
a movie using a 15 Mbit/s ...
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Evaluate the definite integral $\int_0^\infty \frac{W{(x)}}{xe^x}\,dx$
It is a well known identity that
$$
\int_0^\infty \frac{\log{(x)}}{e^x}\,d{x} = -\gamma
$$
This shows how the Euler–Mascheroni constant is directly connected to the exponential function and its ...
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Creating an inequality for the numerator significance
An open mathematical problem is if the Euler-Mascheroni constant is irrational and if so transcendental. There has been some progress in (dis)proving this. It is known that:
$\gamma\in\mathbb{Q}\...
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2
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How to show that $\frac{d}{dx}\sum_{n=1}^\infty\left(\frac xn-\ln\left(1+\frac xn\right)\right)=H_x$
So I was trying to find a series expansion of $\Gamma(x+1)$ (which is the analytic continuation of $x!$ when I bumped into this sum $$\lambda(x)=\sum_{n=1}^\infty\left(\frac xn-\ln\left(1+\frac xn\...
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Expressions for $\sum_{k=1}^\infty\frac{(-1)^k}{k\cdot k!}$
It is known that $$\delta=-e\left(\gamma+\sum_{k=1}^\infty\frac{(-1)^k}{k\cdot k!}\right)$$Where $\delta$ is the Euler-Gompertz constant and $\gamma$ is the Euler-Mascheroni constant. I want to find ...
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Which functions besides $\ln{x}$ make $ \lim \limits_{N \to \infty} \sum_{n=1}^N f'(n) -f(N) $ converge?
Using a very hand-wavy argument, I convinced myself that if, instead of $f(x)=\ln{x}$, we let $f(x)=\sqrt{x}$, we should still get something finite and small. Wasn't really sure where to start to ...
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Other closed forms of $\lim_{k\rightarrow\infty}\left(\sum_{i=0}^k\frac{1}{2i+1}-\sum_{i=1}^k\frac{1}{2i}\right)$
It is known that
$$\lim_{k\rightarrow\infty}\left(\sum_{i=0}^k\frac{1}{2i+1}-\sum_{i=1}^k\frac{1}{2i}\right)=\beta$$converges. I wonder if there are any other closed forms for this limit. At first I ...
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Another strange limit which seems to converges to $\gamma$
Hi in trying to show that the Euler Mascheroni constant is irrational or not I find empiricaly :
$$\lim_{n\to\infty,x\to 0}f_n(x)=\lim_{n\to\infty,x\to 0}\frac{x!!!...!^{x!!...!^{x!...!^{...^{x!}}}}-x!...
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Given that $\lim_{z\rightarrow0}az-\frac{\ln z}{z!}=\gamma$, find the value of $a$ that gives the quickest convergence
You could prove that $$\lim_{z\rightarrow0}az-\frac{\ln (z!)}{z}=\gamma$$For any $a\in\mathbb{R}$. Where $z!$ is the factorial extended to all real numbers (except negative integers). Which value ...
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Sum versus product on Primes
Disclaimer: I had a hard time choosing the title.
Hello MSE!
So in this post I was wondering why $e^\gamma$ has importance in number theory while $\gamma$ and $e$ don't. But I have another question ...
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2
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Is it always true that $\sum\limits_{j=1}^n\frac{1}{j} > \int_1^{n+1} \frac{dt}t$?
This inequality was presented as obvious in this answer. I was interested in understanding the details for this comparison so I started reading up the Euler-MacLaurin formula but this actually ...
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Is this substitution valid?
So I made this identity for Euler's constant $\gamma$:$$\int_{-\infty}^\infty e^{-(e^t+t)}tdt=-\gamma$$But this isn't right for some reason. So what's wrong with my proof?:
$$\int_0^\infty e^{-t}\ln ...
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answer
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Euler's constant's role in the Prime numbers
I know that: $$\lim_{k\rightarrow\infty}\frac{1}{\log p_n}\prod_{i=1}^k\frac{p_i}{p_i-1}=e^\gamma$$Where $p_n$ is the $n$th prime and $\gamma$ is Euler's constant. But why does this make the number $\...
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Prove that the limit converges to $\gamma$
Let $$H(x)=\int_0^1\frac{t^x-1}{t-1}dt$$be the harmonic series and let $$s(x)=\int^\infty_0e^{-t}\ln(t+x)dt$$How do I prove that their difference converges? It seems to me that they approach $\gamma$ (...
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Proof of $\int_0^1\ln\left(\frac{1}{\ln \frac{1}{x}}\right)dx=\gamma$
After solving this problem I found out that: $$\int_0^1\ln^sxdx=(-1)^ss!$$Which can be proven by induction. Anyways, I realized that I could use this to prove that $$-\int_0^1\ln\left(\ln \frac{1}{x}\...
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How to prove that $\lim_{k\rightarrow\infty} H_k-\log_b(x)$ converges/diverges?
It is known that: $$\lim_{k\rightarrow\infty} H_k-\log(k)=\gamma$$Where $\gamma$ is Euler's constant. What if we change the base from $e$ to any other number $b$? How do we prove that this converges/...
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Approximations of Euler's constant using $\pi$
Disclaimer: This is for recreational purposes.
Hello MSE! So while my research paper about Euler's constant $\gamma$, I created this amazing approximation for it: $$\frac{\pi^2}{12000}-\ln((10^{-3})!)*...
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Integrals of the form $\int_0^1\left(\frac{1}{x+1}-\ln\left|\frac{x+1}{2}\right|-\ln\ln\left|\frac{x+1}{x-1}\right|\right)dx$
How could we evaluate the following integrals? (Integrands pictures above)
Blue:
\begin{align}
\int_0^1 \left(\frac{1}{x+1}-\ln\left|\frac{x+1}{2}\right|-\ln\ln\left|\frac{x+1}{x-1}\right|\right)dx &...
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Prove that $\lim_{x\rightarrow\infty}\sum_{k=0}^x\frac{1}{2k+1}-\sum_{k=1}^x\frac{1}{2k}$ converges
I thought up of the limit in the title by playing around with the limit definition of Euler's constant. I think I was able to prove that it is convergent. Is the following proof correct?
Let $H_e(x)=\...
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How to prove that $\lim_{k\rightarrow\infty}\ln k - \frac{1}{2}H(k)$ diverges?
It is known that $$\lim_{k\rightarrow\infty}\ln k - H(k)=\gamma$$(Where $H(k)$ is the $k$th harmonic number and $\gamma$ is Euler's constant). I decided to put a random $\frac{1}{2}$ in the front of $...
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Where does Euler write the definition $\gamma=\sum^\infty_{k=1}\left(\frac1k-\ln\left(1+\frac1k\right)\right)$ of the Euler-Mascheroni Constant?
I am writing a research paper on the Euler-Mascheroni Constant when I mentioned this sum: $$\gamma=\sum^\infty_{k=1}\left(\frac{1}{k}-\ln\left(1+\frac{1}{k}\right)\right)$$ which was derived by Euler. ...
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An interesting integral about γ [closed]
It is well known that
$$\gamma = \lim_{n \to +\infty} (H_n − \ln(n)) = 0.5772(...)$$
where $H_n$ is the sum of reciprocals of all integers from $1$ to $n$.
Prove that $$\int_1^\infty \frac{\{ x\}}{x^...
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Curiosities of the function $Q(x)=\sum_{n=1}^\infty \frac{P_n(x)}{n(2n+1)}$ where $P_n(x)$ is a sequence of all polynomials with unit coefficients
Backround:
I have been studying the peculiar function $$Q(x)=\sum_{n=1}^\infty \frac{P_n(x)}{n(2n+1)}$$
where $P_n(x)$ is the set of all polynomials with unit coefficients, defined by the binary ...
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Construction, similar to Chow's EL-numbers? Is it valid? What are the properties?
The idea of EL-numbers, proposed by Chow, impressed me very much, so I decided to build something similar and look what this will turn out.
Instead of $\exp(x)$ and $\ln(x)$ functions as the building ...
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A constant written in terms of non-elementary functions
I found out that: $$\frac{\int \Pi(x)H(x)dx-\Pi(x)}{\int \Pi(x)dx}=\gamma$$ Where $H(x)$ is the $x$th harmonic number, $\Pi(x)$ is the analytic continuation of $z!$, and $\gamma$ is the Euler-...
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Why rate of convergence is studied?
There are lots of papers on the Internet about sequences with different rates of convergence towards Euler's constant and every year more are published for better rates of convergence. Why is having ...
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How did Euler calculate his $\gamma$ constant to 16 digits? [closed]
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?
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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$$\operatorname{li}(x):=\int_0^x\frac{dt}{\...
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Identity for the Euler-Mascheroni constant
Euler created the following expression for the Euler-Mascheroni constant: $$\gamma=\sum^\infty_{k=1} \left(\frac{1}{k}-\ln\left(1+\frac{1}{k} \right)\right)$$ This converges more rapidly than the ...
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How do I prove this Euler-Mascheroni limit?
There is a number called the Euler-Mascheroni constant that is defined as the limiting difference between the harmonic series and the natural logarithm. In other words:
$$
\gamma =
\lim_{N \to \infty}\...
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votes
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answer
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Deriving $\gamma \approx H(n)-\ln(n+1)+\frac{1}{2(n+1)}+\frac{1}{12(n+1)^2}$
The Euler-Mascheroni constant can be represented geometrically by the infinite sum of the areas in blue in the following picture, which is the area between the curve $y=1/x$ and the harmonic numbers.
...
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Show without a computer that : $\frac{\gamma}{2-\gamma}\pi-\ln(3+\gamma)<0$ where $\gamma$ is the Euler-Mascheroni constant
Challenging problem :
Show that :
$$\frac{\gamma}{2-\gamma}\pi-\ln(3+\gamma)<0$$
I come up with this inequality with What is the limit $\lim_{x\to 0}\left(\frac{x!x!!!x!!!!!...}{x!!x!!!!x!!!!!!...}\...
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Euler-Mascheroni Constant and the Digamma Function
I have been trying to prove without success that the digamma function evaluated at $1$ is equal to the Euler-Mascheroni constant $\gamma$.
I would like to know if there is any way of doing it by just ...
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Is there any intuition of why the both, regularized logarithm of zero is $-\gamma$ and the regularized logarithm of Bernoulli umbra is $-\gamma$?
If we take the MacLaurin series for $\ln(x+1)$ and evaluate it at $x=-1$, we will get the Harmonic series with the opposite sign: $-\sum_{k=1}^\infty \frac1x$. Since the regularized sum of the ...
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Elementary proof for $\varphi(n)\ge\frac n{2\log(\log(n))}$
Trying to derive an elementary bound for Euler totient function $\varphi(n)$ to be $\mathcal{O}(\frac n{\log(\log(n))})$, I thought to prove a weak version of the well-known inequality
$${\...
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A series on Stieltjes constants, $\sum_{n=0}^\infty \frac{\gamma_n}{n!}$, where is the mistake?
Stieltjes constants are defined as:
$$\gamma_n=\lim_{m\to\infty}\left[ \left( \sum_{k=1}^m\frac{(\ln k)^n}{k}\right)-\frac{(\ln m)^{n+1}}{n+1} \right]$$
I want to compute the following series:
$$\...
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5
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Euler-Mascheroni constant in Bessel function integral
I am currently juggling some integrals. In a physics textbook, Chaikin-Lubensky [1], Chapter 6, (6.1.26), I came upon an integral that goes
\begin{equation}
\int_0^{1} \textrm{d} y\, \frac{1 - J_0(y)}{...
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1
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How to find an equivalent of $\int_{1}^{+\infty}\exp{(-x^n)}dx$?
It's an exercise, I need to find an equivalent of :
$$\int_{1}^{+\infty}\exp{(-x^n)}dx$$
I tried this :
Let $x=u^{1/n}.$ Then $dx=\frac{1}{n}u^{1/n-1}\,du,$ so :
\begin{align}
\int_1^{+\infty} e^{-x^n}...
1
vote
1
answer
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Generalization of $\gamma$ as limit involving primes
The following limit is utilized in Merten’s theorem.
$$
\gamma = \lim\limits_{n\to\infty}\left(-\log\log n - \sum_{p\text{ prime}}^n\log\left(1-\frac{1}{p}\right)\right)
$$
I’m interested in the ...
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Evaluation of $\sum_{n=1}^{\infty}\frac{1}{n(n!)}$
WolframAlpha gives the evaluation as $\text{Ei}(1)-\gamma$. It does not offer a step-by-step solution. Where does this come from?
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Where does the Euler Mascheroni constant come from in the formula $\sum_{n≤x} 1/n= \log(x)+\gamma +O(1/x)$
From this question,
Asymptotic formula for $\sum_{n\le x}\frac 1n$
The author claims without proof that this identity holds.
$$\sum_{n≤x} 1/n= \log(x)+\gamma +O(1/x) .$$
It is apparently well known ...
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votes
1
answer
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How to show that Euler constant lies in $(0,1)$ [duplicate]
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 ...
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votes
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answer
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A sum that's possibly equal to the Euler-Mascheroni Constant $\sum_{n=1}^\infty \frac{\ln n!}{n^3}$
The following interesting sum seems to approach the Euler-Mascheroni constant $\gamma$.
$$\sum_{n=1}^\infty \frac{\ln n!}{n^3} \overset{?}{=} \gamma$$
I've looked at the different ways to express the ...
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2
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547
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Is $\sum_{j=1}^\infty\sum_{n=1}^\infty\left(\frac{e^{-j/n}}{n^2}-\frac{e^{-n/j}}{j^2}\right)=\gamma ?$
A friend proposed the following problem:
$$\sum_{j=1}^\infty\sum_{n=1}^\infty\left(\frac{e^{-j/n}}{n^2}-\frac{e^{-n/j}}{j^2}\right)=\gamma,$$
where $\gamma$ is the Euler-Mascheroni constant.
The ...
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Valid proof that Euler's Constant $\gamma$ is between $0$ and $1$?
I was wondering if there is any reasonable way/theory to do calculations with divergent limits of a sequence. I was trying to prove that Euler's constant $$\gamma = \displaystyle{\lim_{n \to \infty}}
\...
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How can I continue my proof that the difference of cosine integrals is the Euler Mascheroni constant?
$\newcommand{\d}{\,\mathrm{d}}\newcommand{\cn}{\mathrm{Cin}}\newcommand{\ci}{\mathrm{Ci}}$I am very new to the Ci, Si functions - I saw these an hour ago! In particular, I saw an identity involving ...
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Book suggestions to learn more about Euler-Mascheroni constant
I want to learn more about the Euler-Mascheroni constant, $\gamma$ and do research on a problem related to it. The problem does not necessarily has to be the question of rationality/irrationality of $\...
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Triple product $\displaystyle\prod_{k=1}^{\infty} \prod_{n=1}^{\infty} \prod_{m=1}^{\infty}(k+n+m)^{\frac{(-1)^{k+m+n}}{k+m+n}}$
Prove that
$$\prod_{k=1}^{\infty} \prod_{n=1}^{\infty} \prod_{m=1}^{\infty}(k+n+m)^{\frac{(-1)^{k+m+n}}{k+m+n}}=\frac{\mathbb{A}^{\frac{3}{2}}}{\pi^{\frac{3}{4}} e^{\frac{1}{8}-(\frac{7}{12}+\gamma) \...
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$\frac {\zeta(2)}{e^2} + \frac {\zeta(3)}{e^3} + \frac {\zeta(4)}{e^4} + \frac {\zeta(5)}{e^5}.....?$
$$\frac {\zeta(2)}{e^2} + \frac {\zeta(3)}{e^3} + \frac {\zeta(4)}{e^4} + \frac {\zeta(5)}{e^5}.....?$$
I tried to solve it by using this product formula,
$$\frac 1{\Gamma (x)}=xe^{\gamma x} \prod_{n=...
user988316
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2
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Proof that the limit of a series of exponential distributed random variables follows a Gumbel distribution
How can I prove that given $X_k \sim \text{Exp}(1/k),$
$$\lim_{n\to\infty}\left(-\log(n) + \sum_{k=1}^n X_k \right)\sim \text{Gumbel}(\mu =0,\beta = 1)$$
?
I see that $\log(n)$ is the integral of $1/x$...
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