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

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10
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2answers
116 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= ...
6
votes
1answer
67 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 \int_0^{1} \!\cfrac 1 {1 + \cfrac 1 {1 + \cfrac 1 {\ddots + \cfrac 1 { 1 + \psi ...
1
vote
2answers
51 views

Must square root of e be positive?

I have always thought that there is two solutions to the square root of a real number, one being positive and the other being negative. However, in Penrose's book, A Road to Reality, he seems to claim ...
1
vote
1answer
47 views

Possible values of $\lim_{x \rightarrow \infty} \left(\frac{p(x)}{q(x)}\right)^x$

I was asked by some freshmen the follwing two questions regarding the limit: Using the fact that $$\lim_{x \rightarrow \infty} \left(1 + \frac{1}{x}\right)^x = e$$ evaluate $$\lim_{x \rightarrow ...
0
votes
1answer
24 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) ...
1
vote
2answers
60 views

By expanding $e^x$ into a series prove the following inequality

By expanding $e^x$ into a series $\sum e^x$ prove that $$\forall x \in \mathbb{R}, x \ge 0 \implies e^{x-1} \ge x$$ Also show when this inequality becomes equality. I'm not really sure how to attack ...
1
vote
1answer
23 views

possible meaning of a constant

I am sorry if the question is ambiguous. Is there any result closely related to the constant $\frac{1+\sqrt{2}}{2}$? It does not count if you say " it is a root of $x^2-x-1/4=0$".
0
votes
2answers
60 views

Inequality Regarding e

A proof done in class called upon this inequality: $$\left(1+\frac{1}{n}\right)^n\lt e\lt\left(1+\frac{1}{n}\right)^{n+1}$$ How can this inequality be proven?
0
votes
4answers
92 views

Proof that $ 3 > (1+\frac{1}{n})^n \geq 2$

I am studying computer science in first term, and i got a task that i was not able to solve for a long time now. I have to prove that $ 3 > (1+\frac{1}{n})^n>=2$ for every $n \in ...
2
votes
1answer
20 views

Statistics: How to use Poisson to determine if event would not occur?

This is a follow up to my previous question disk manufacturer averages 0.2 missing pulses per disk. let X denote # of missing pulses I am using poisson distribution I'm having trouble determining ...
1
vote
3answers
57 views

e as sum of an infinite series

I read that $e = \sum_{i=0}^\infty$$ 1\over n!$. This isn't immediately obvious to me, and I can't find proof of this. Can somebody explain to me, how do I prove this from definition $e = \lim_{n\to ...
0
votes
1answer
53 views

How to prove this equality about e

I have the assumption that the following holds: $$\lim_{n \to \infty}\frac{1}{n^2} \cdot \sum_{i = 0}^n \left(1 - \frac 1n \right)^i = 1 - \frac{2}{e}.$$ However, I am totally not sure about it. How ...
0
votes
0answers
69 views

Consequences of irrationality of e

We know that $e$, $\pi$ are irrational... But WHY do we know it? I am going to give a lecture about irrationality of e, and I'm looking for a reason why it is an interesting subject. I know the proof, ...
6
votes
6answers
236 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} ...
9
votes
3answers
153 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
73 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 ...
0
votes
1answer
38 views

How to find the inverse function of Euler's number?

Given: $f(x)= \dfrac{e^x}{1+9e^x}$ , what steps would I take to find its inverse? I tried following the steps on finding the inverse of a normal function but I keep getting one of the variables to ...
0
votes
1answer
31 views

Specifying variables in Maple

Two questions 1)How do you express Euler's number (≈2.718) in Maple. For example I want ln(e) to simplify to 1. 2)How do you have a constant with a subscript? For example if I wanted to input the ...
26
votes
6answers
3k views

What Gauss *could* have meant?

I was reading the Wikipedia entry on Euler's identity ($e^{i\pi}+1=0$) and I came across this statement: "The mathematician Carl Friedrich Gauss was reported to have commented that if this formula ...
11
votes
3answers
499 views

Very challenging: max{floor,ceil}=?

I spotted a pattern while trying to generalize a problem. (EDIT: said problem has been removed from this post to avoid confusion. EDIT(2): Here is the problem again: ...
4
votes
1answer
92 views

Intuitive reason for why the Gaussian integral converges to the square root of pi?

This is a very famous problem, which is commonly taught when students begin learning about multivariable integration in polar coordinates. However, it has always bothered me that we recieved such an ...
3
votes
1answer
91 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 ...
9
votes
1answer
142 views

An integral with $e^{1+e^x}$ I had trouble working through

I had an analysis test earlier this morning and came across this integral, which I couldn't figure out. Parts of it are easy, but after integrating $y$ you're left integrating $xe^{1+e^x}$ which had ...
1
vote
1answer
94 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 ...
35
votes
2answers
894 views

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

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_{1}x_{2} \cdots ...
13
votes
1answer
281 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
55 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 ...
0
votes
2answers
71 views

For which positive numbers $a$ is it true that $a^x \ge 1 + x$ for all $x$?

For which positive numbers $a$ is it true that $a^x \ge 1 + x$ for all $x$? This is question 24, chapter 3 from Stewart's Early Transcendental problems plus. I think I'm on the right track, but I'm ...
0
votes
1answer
69 views

Function relating Euler's constant and the golden ratio

Okay, I was messing around on Excel with some coefficients and I stumbled onto this. Not sure if it converges but it gets pretty damn close around the 1024th term mark. Was wondering if somebody could ...
2
votes
2answers
37 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
181 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 ...
4
votes
2answers
68 views

Is this a legal transformation?

To be found: $$\lim \left(1+\frac{2}{n}\right)^n$$ Presuppose $~~\lim \left(1+\frac{1}{n}\right)^n=e~~$ is already shown. Expanding the first equation: $$\lim ...
3
votes
4answers
185 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 ...
0
votes
1answer
52 views

Upper bound of natural logarithm

I was playing looking for a good upper bound of natural logarithm and I found that $$\ln x \le x^{1/e}$$ apparently works: Can someone give me a formal proof of this inequality?
1
vote
1answer
53 views

Rewriting in $y=A_0\cdot e^{at}$

How do you rewrite $y = −8(1.589)^{t − 3}$ in $y=A_0e^{at}$ form for appropriate constants $A_0$ and $a?$ For other problems I took the $\ln$ of the number inside the parenthesis. So for example I ...
1
vote
2answers
76 views

Limit of ${x}^{\frac1{1-x}}$ as $x$ approaches 1

I have the following homework question, and I've been stuck on it for the last 2 hours. I have no idea how to find the answer. $$\lim\limits_{x\to1} {x}^{\frac1{1-x}}$$ I have tried manipulating the ...
0
votes
2answers
54 views

How to isolate a variable when its on both sides of equation

$ ex-1=x $ How do rearrange this equation to isolate x i.e in the form of x=.....
3
votes
1answer
69 views

Why is $\int |e^{ix}|^2 dx = x + C$?

Quick question: Wolfram Alpha tells me that $$\int |e^{ix}|^2 dx = x + C$$ Why is that?
2
votes
0answers
43 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 ...
12
votes
1answer
672 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
80 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 ...
0
votes
1answer
66 views

Dynamic Sizing of Circles Along a Logarithmic Spiral

I have created an logarithmic spiral in HTML canvas, and plotted circles along it. Using your mouse scroll wheel you can zoom in and out of the spiral (which works) – but I am having problems updating ...
3
votes
3answers
107 views

definition of the constant $e$

To my knowledge there are two possible ways to define $e^x$ $$e^x = \sum_{i=0}^{\infty}\frac{x^i}{i!}$$ $$e^x = \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n$$ So my question is: Why does… ...
1
vote
1answer
38 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
60 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
65 views

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

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 ...
0
votes
2answers
45 views

How to plot a curve in complex plane that include e constant

I can understand how a complex numbers such as $10 + 7i$ can be plotted in complex plane with Imaginary and Real axis. But I have no idea how to approach this problem. I have been asked to plot ...
1
vote
0answers
39 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
142 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
2answers
68 views

Evaluate $a^i$ where $a$ is real:

I have a question involving the evaluation of $3^i$, but I am unsure how to do this. I know how to solve such questions involving $e^{i\theta}$, but how does this work with a different base? (I ...