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

learn more… | top users | synonyms

0
votes
3answers
71 views

Is this : $\lim \sup\frac{\sigma(n)}{n} , n \to\infty $ has a finite limit?

The asymptotic growth rate of the sigma function can be expressed by : $$\lim \sup\frac{\sigma(n)}{n\log(\log(n))}=e^{\gamma}$$ $$n \to\infty$$ according to the above limit , Is this : $$\lim ...
1
vote
1answer
35 views

What Does The Sequence $a_ N = \frac{\sum_{j=1}^{N} \{\frac{N}{j}\} }{N}$ Converge to?

In some of my number theoretical calculations I saw the sequence $$a_ N = \frac{\sum_{j=1}^{N} \{\frac{N}{j}\} }{N}$$ for $1\leq N$ and $\{x\}$ means the fractional part of $x$. I checked with a ...
1
vote
1answer
27 views

Find the region R for which the sequence converges

Find the region $(x,y) \in R$ for which the following sequence converges $$\left| e^n\frac{(\sqrt{y}-\sqrt{x})^{2n}}{x^n} \right| \to 0$$ I am currently doing number theory research on studying the ...
0
votes
1answer
37 views

Solve the differential equation using separation of variables: $\frac{dy}{dx} = e^{3x+2y}$

I need to solve using the method of separation of variables. I got to this point but I'm not sure if I'm on the right track: $$\frac{-1}{2e^{2y}} + C = \frac{e^{3x}}{3} + C$$ It looks like now I ...
0
votes
2answers
52 views

L'Hôpital's rule exercise with sqrt(x) as exponent

I'm a bit stuck trying to find the limit of the following function: $$\lim_{x \to 0^+}\,\,{x^{\sqrt{x}}} $$ We are expected to use L'Hôpital's rule, and thus far I've managed to resolve the equation ...
0
votes
1answer
28 views

Solving a variable in a matrix equation?

I am having trouble solving for a in the problem below. I've simplified it down to: $e^{14} = ln(e^e \cdot a)$. I'm not really sure where to go from here.
2
votes
2answers
62 views

How to solve the derivative of $b^x$ using the defintion

I know that the derivative of $b^x$ is just $b^x \log{(b)}$, and I've seen it being derived using chain rule and such (not that I understand how it's done, I just learned about $e$ today so using the ...
16
votes
8answers
439 views

Explain why $e^{i\pi} = -1$ to an $8^{th}$ grader?

I am an $8^{th}$ grader that is taking Algebra I. But nearly everyday I try to learn things outside of what I am learning in class. Quite a while ago I discovered that $e^{i\pi} = -1$. This ...
1
vote
1answer
94 views

How is de = 1 (mod ϕ(n)) calculated

I am reading RSA algorithm. So, I was writing a question but I saw this question and still couldn't understand it. If $$e\cdot d \equiv 1 \pmod{\varphi(n)},$$ then $$ed=k\cdot \varphi(n)+1, \qquad ...
2
votes
3answers
98 views

Is it possible to intuitively explain, how the three irrational numbers $e$, $i$ and $\pi$ are related?

I read a bit about this equation: $e^{i\pi}=-1$ For someone knowing high school maths this perplexes me. How are these three irrational numbers so seemingly smoothly related to one another? Can this ...
4
votes
5answers
122 views

Graph of the function $x^y = y^x$, and $e$ (Euler's number).

Earlier, I was using the Desmos Graphing Calculator, and I wanted to remind myself of what the graph of the function $x^y = y^x$ looked like. If you have never seen what it looks like before, it is ...
7
votes
2answers
236 views

Why does $\int_0^{\infty}\frac{\ln (1+x)}{\ln^2 (x)+\pi^2}\frac{dx}{x^2}$ give the Euler-Mascheroni constant?

I'd like to see the reason why $$\int_{0}^{\infty}\frac{\mathrm{ln}(1+x)}{\mathrm{ln}^2(x)+\pi^2}\frac{dx}{x^2}=\gamma$$ where $\gamma$ is the Euler-Mascheroni constant. I don't have any 'neat ...
0
votes
1answer
32 views

Justify the steps of the integral representations of the Euler-Mascheroni constant $\gamma$

These are the steps in order to get the Euler-Mascheroni constant $\gamma$ there were steps before where I'm starting but I didn't have questions about the justifications of those so I didn't include ...
1
vote
2answers
79 views

Approximating the value of Euler's constant?

I'm asked the following: Using the series that defines $\gamma$, Euler's constant, what's the minimum number of terms that we have to sum in order to calculate $\gamma$ with an error less than $2 ...
0
votes
2answers
53 views

How to show that $\lim_{x\rightarrow \infty }(1+\frac{x}{n})^{n}=e^{x}$ [duplicate]

I have a problem: $E(x)=e^x$, $L(x)=\ln (x)$, $E^{-1}(x)=L(x)$. Show that $\lim_{n\rightarrow \infty }(1+\frac{x}{n})^{n}=e^{x}$ Hint: use $f(t)=\ln (1+xt)$ and look at $f'(0), x\neq 0$. I ...
1
vote
0answers
30 views

Sum of random numbers and e constant [duplicate]

I recently came across this interesting fact: Take some (pseudo)random numbers between 0 and 1. Now sum this and count how many you need in order for their sum to be greater than 1. If you repeat ...
3
votes
1answer
104 views

Euler Mascheroni Constant is Zero

$$ H_n - \ln n = \int_0^1 \frac{1 - x^n}{1 - x} dx - \int_1^n \frac{dy}{y} $$ Let $x = \frac{y - 1}{n - 1}$, or $y = (n-1)x + 1$. Then, $dy = (n - 1) dx$. $$ H_n - \ln n = \int_0^1 \frac{1 - x^n}{1 ...
2
votes
2answers
99 views

Unusual integral

I got a clock as a gift recently. It has a very novel face in that the hour positions are given by a complex formula. For the most part, I have been able to verify the calculations presented as ...
4
votes
2answers
101 views

Evaluate the double integral $\int _0^1\int _0^1\frac{x+i}{(1-ix y) \ln (x y)} \,dx\,dy$

We know that $$\int _0^1\int _0^1\frac{x-1}{(1+x y) \ln (x y)} \, dx\,dy=\gamma$$ $$\int _0^1\int _0^1\frac{x+1}{(1-x y) \ln (x y)}\,dx\,dy=\ln \frac4\pi$$ I wonder what would be $$\int ...
2
votes
1answer
41 views

Does Euler-Mascheroni constant belong to the ring of periods?

I wonder whether $\gamma$ belongs to the ring of periods?
2
votes
1answer
59 views

Does $ \lim_{n \to \infty} \frac{\operatorname{exp}(H_n)}{n+1} $ exist? [closed]

Question: Does $ \lim_{n \to \infty} \frac{\operatorname{exp}(H_n)}{n+1} $ exist? If so, what is its value? I know that the answer to the second part is $e^\gamma$, where $\gamma$ is the ...
12
votes
2answers
152 views

Showing $\gamma < \sqrt{1/3}$ without a computer

In 1735 Euler gave the value of $\gamma$ as $0.577218.$ The constant is generally defined as the limit of the difference between the harmonic series and $\log n:~\gamma= ...
12
votes
1answer
208 views

A fractional part integral giving $\frac{F_{n-1}}{F_n}-\frac{(-1)^n}{F_n^2}\ln\left(\!\frac{F_{n+2}-F_n\gamma}{F_{n+1}-F_n\gamma}\right)$

I've been asked to elaborate on the following evaluation: $$ \begin{align}\\ \displaystyle {\large\int_0^{1}} \!\cfrac 1 {1 + \cfrac 1 {1 + \cfrac 1 {\ddots + \cfrac 1 { 1 + \psi ...
0
votes
1answer
43 views

Euler homogeneous equations

Find the general solution of the Euler homogeneous equations? dy/dx= (2y-x)/(2x-y) Using the substitution y=vx v+x(dv/dx)= (2xv-x)/(2x-xv) v+ x(dv/dx)= x(2v-1)/x(2-v) v+ x(dv/dx)= (2v-1)/(2-v) ...
6
votes
6answers
416 views

Finding $\sum_{k=1}^{\infty} \left[\frac{1}{2k}-\log \left(1+\frac{1}{2k}\right)\right]$

How do we find $$S=\sum_{k=1}^{\infty} \left[\frac{1}{2k} -\log\left(1+\dfrac{1}{2k}\right)\right]$$ I know that $\displaystyle\sum_{k=1}^{\infty} \left[\frac{1}{k} ...
10
votes
3answers
191 views

Possible new definition of Gamma (Euler-Mascheroni Constant): $\lim_{x \to 0} (-\ln ( \sqrt[x]{x!} )) = \gamma$

I think I've discovered a new definition for the Euler-Mascheroni Constant (Gamma) I can't find it online anywhere, has anyone seen it before? $$\lim_{x \to 0} (-\ln ( \sqrt[x]{x!} )) = \gamma$$
3
votes
1answer
82 views

Infinite product: $(1-0.5^2)(1-0.5^3)(1-0.5^4)…$

Find a closed form for the value of the infinite product $(1-0.5^2)(1-0.5^3)(1-0.5^4)...$ I know it converges. At first I thought it was the Euler–Mascheroni constant, but it's only accurate to about ...
3
votes
1answer
101 views

Asymptotics of coefficients $[x^n] \frac{1}{\Gamma(1+x)}$ as $n$ is great

I am interested in the behaviour, as $n$ is great, of the coefficients $g_n$ in the Maclauren expansion of $\displaystyle \frac{1}{\Gamma(1+x)} $. We have $$ \frac{1}{\Gamma(1+x)}=\sum_{n=0}^\infty ...
1
vote
1answer
102 views

An integral representation for $\psi$

Let $\displaystyle \gamma$ denote the Euler constant defined by $\displaystyle \gamma := \lim\limits_{n \to \infty} \left(\frac11+\frac12+\cdots+\frac1n- \log n\right)$. Here is an integral for ...
41
votes
2answers
1k views

Evaluating $\int_{0}^{1}\cdots\int_{0}^{1}\bigl\{\frac{1}{x_{1}\cdots x_{n}}\bigr\}^{2}\:\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}$

Here is my source of inspiration for this question. I suggest to evaluate the following new one. $$ I_{n}:= \int_0^1 \! \cdots \! \int_0^1 \left\{\frac{1}{x_1x_2 \cdots x_n}\right\}^{2} ...
14
votes
1answer
411 views

A closed form for the series $\sum_{n=1}^{\infty} \frac{H_n^2-(\gamma + \ln n)^2}{n}$

I have found a closed form for the following new series involving non-linear harmonic numbers. Proposition. $$\sum_{n=1}^{\infty} \dfrac{H_n^2-(\gamma + \ln n)^2}{n} = ...
0
votes
0answers
82 views

Step in proof involving Euler-Mascheroni constant

I was just looking at a proof that shows how the Euler-Mascheroni constant exists and is situated between 0 and 1 . However, I stumbled across a step in the proof that doesn't seem very obvious to me ...
2
votes
2answers
42 views

Zeta Function $\zeta(1\pm1/n)$ and Euler's constant.

How do I show that $$\lim_{n\to\infty}{\zeta(1+1/n)+\zeta(1-1/n)}=2\gamma$$ and $$\lim_{n\to\infty}{\zeta(1+1/n)-\zeta(1-1/n)}=\infty,$$ where $\gamma$ is the Euler's constant?
2
votes
2answers
204 views

Why is $1 + \frac{1}{2} + \frac{1}{3} + … + \frac{1}{n} \approx \ln(n) + \gamma$?

On StackExchange, I read that the harmonic series up to $\frac{1}{n}$ is approximately $\ln(n) + \gamma$, where $\gamma$ is the Euler-Mascheroni constant, which is close to $0.5772$. When I researched ...
3
votes
4answers
194 views

Compute $\sum (1+\frac12+\dotsb+\frac1n-\ln (n+\frac12)-\gamma)$

Compute $$\sum_{n=1}^\infty \Big(1+\frac12+\dotsb+\frac1n-\ln (n+\frac12)-\gamma\Big)$$ where $\gamma$ is Euler's constant It seems to be difficult, I have no idea go get started Thank you very ...
2
votes
0answers
72 views

Mertens Constant and Euler–Mascheroni constant

I found this titillating equation: $$M = \gamma + \sum_{p} \left[ \ln\! \left( 1 - \frac{1}{p} \right) + \frac{1}{p} \right]$$ where $\displaystyle M=\lim_{n \rightarrow \infty } \left( \sum_{p\,\leq ...
13
votes
1answer
756 views

Derivative of Riemann zeta, is this inequality true?

Is the following inequality true? $$\gamma -\frac{\zeta ''(-2\;n)}{2 \zeta '(-2\;n)} > \log (n)-\gamma$$ This for $n$ a positive integer, $n=1,2,3,4,5,...$, and more precisely when $n$ approaches ...
2
votes
1answer
109 views

Pole of Riemann zeta and Riemann zeta zeros, prove this relation.

Prove this relation: $$\displaystyle \lim_{s\to 1} \, \left(\zeta (s)-\frac{\zeta '(s-1+\rho _n)}{\zeta \left(s-1+\rho _n\right)}\right)=\gamma -\frac{\zeta ''(\rho _n)}{2 \zeta '(\rho ...
1
vote
1answer
43 views

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.
0
votes
1answer
80 views

How do i prove the Euler-Mascheroni constant is positive?

Define $\gamma=\lim_{n\to\infty} \sum_{k=1}^n 1/k - ln(n)$. I know that $\gamma$ is nonnegative, but i don't know how to prove that it is positive.
0
votes
0answers
77 views

How to show that $-\int_0^\infty{\ln(x)\exp(-x)dx}=\gamma$ where $\gamma$ is the Euler's constant. [duplicate]

I know that the Euler's constant $\gamma$ is defined by the term $\lim\limits_{N\rightarrow\infty}{\left(\sum_{n=1}^{N}{\frac{1}{n}}-\ln N\right)}$. But I can't see how this term is equal to the ...
1
vote
0answers
49 views

About Abel Summation

http://arxiv.org/pdf/math/0504289v3.pdf Here i'm trying to understand page 5. Writer uses the abel sum to find the sum of the prime's reciprocals. So he founds the formula (2.2.1) Now here y=2 ...
2
votes
2answers
175 views

Euler-Mascheroni constant: understanding why $\lim_{m\rightarrow \infty} \sum_{n=1}^{m} (\ln (1 + \frac{1}{n})-\frac{1}{n+1})= 1 - \gamma$

I am trying to understand why the Euler-Mascheroni constant $\displaystyle \gamma = \lim_{n \rightarrow \infty} \left ( \sum_{k=1}^n \frac{1}{k} - \ln n \right )$ is equal to $1 - \displaystyle ...
2
votes
1answer
45 views

An Euler-Mascheroni-like sequence [duplicate]

How does one compute the limit of the sequence: $$\sum_{k = 0}^{n}\frac{1}{3k+1} - \frac{\ln(n)}{3}$$ I would apreciate a hint.
0
votes
1answer
347 views

What is the limit of difference between harmonic series and natural logarithm of n+1?

I'm an undergraduate student in geology and I'm dealing with a project in math. The last question of the project gives me the harmonic series (An = 1 + 1/2 + ... + 1/n) and this natural logarithm L = ...
4
votes
1answer
2k views

Simple proof Euler–Mascheroni $\gamma$ constant

I'm searching for a really simple and beautiful proof that the sequence $(u_n)_{n \in \mathbb{N}} = \sum\nolimits_{k=1}^n \frac{1}{k} - \log(n)$ converges. At first I want to know if my answer is OK. ...
5
votes
2answers
121 views

Integral $\int_0^{\infty} \log(x) e^{-x^2} \mathrm{d}x = -\frac{1}{4}\sqrt{\pi} (\gamma + \log(4)).$

While trying to compute the expected value $E[\log(X)]$ for a normally distributed variable $X$ I found the following integral $$\int_{0}^{\infty}\log\left(x\right) {\rm e}^{-x^{2}}\,{\rm d}x =-\,{1 ...
16
votes
1answer
356 views

Infinite Series $\sum_{n=1}^{\infty}\frac{(-1)^n}{n}\left\lfloor\frac{\log(n)}{\log(2)}\right\rfloor$

How to prove that $$\sum_{n=1}^{\infty}\frac{(-1)^n}{n}\left\lfloor\frac{\log(n)}{\log(2)}\right\rfloor=\gamma$$ Can we find a known value for ...
8
votes
2answers
127 views

Why is $-\gamma = \int_0^1 \frac{e^{-z}-1}{z}dz+\int_1^\infty \frac{e^{-z}}{z}dz$

It seems like the sum of the two RHS integrals is "well known"$^\dagger$ to be Euler's constant: $$\gamma \equiv \int_1^\infty \frac{1}{\lfloor z\rfloor} - \frac{1}{z}dz \quad\stackrel{?}{=}\quad ...
1
vote
1answer
104 views

Proving this identity $\gamma=1+\ln(\frac{1}{2})+\sum_{k=1}^{\infty}(-1)^{k+1}\frac{\zeta(k+1)-1}{k+1}$ where $\gamma$ is the Euler-Masceroni constant

I've seen this identity here $$ \displaystyle \gamma=1+\ln(\frac{1}{2})+\sum_{k=1}^{\infty}(-1)^{k+1}\dfrac{\zeta(k+1)-1}{k+1} $$ and I'd like to know how it is deduced. Could anyone help? Thanks. ...