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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4
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1answer
65 views

How to prove this series: $\displaystyle \sum_{n=1}^{\infty }\frac{\left ( -1 \right )^{n}\ln n}{n}=\gamma \ln 2-\frac{1}{2}\ln^22$ [duplicate]

How to prove this series $$\sum_{n=1}^{\infty }\frac{\left ( -1 \right )^{n}\ln n}{n}=\gamma \ln 2-\frac{1}{2}\ln^22$$ and \begin{align*} \sum_{n=1}^{\infty }\frac{\left ( -1 \right )^{n}\ln \left ( ...
1
vote
0answers
63 views

Series for Stieltjes constants from $\gamma= \sum_{n=1}^\infty \left(\frac{2}{n}-\sum_{j=n(n-1)+1}^{n(n+1)} \frac{1}{j}\right)$

Euler's constant has the following representations (Euler-Mascheroni constant expression, further simplification, http://math.stackexchange.com/a/129808/134791, Question on Macys formula for ...
2
votes
0answers
183 views

Why is $e$ close to $H_8$, closer to $H_8\left(1+\frac{1}{80^2}\right)$ and even closer to $\gamma+log\left(\frac{17}{2}\right) +\frac{1}{10^3}$?

The eighth harmonic number happens to be close to $e$. $$e\approx2.71(8)$$ $$H_8=\sum_{k=1}^8 \frac{1}{k}=\frac{761}{280}\approx2.71(7)$$ This leads to the almost-integer ...
3
votes
1answer
68 views

Why Can my Phone Calculator do $e^{\pi\sqrt{-1}}$ but not $\sqrt{-1}$?

When I type in the identity $e^{\pi\sqrt{-1}}$ on my phone calculator (LG phone running Android), I get the correct result of $-1$ However, when I simply type $\sqrt{-1}$, it returns an error. Why ...
4
votes
1answer
85 views

What is known about the 'Double log Eulers constant', $\lim_{n \to \infty}{\sum_{k=2}^n\frac{1}{k\ln{k}}-\ln\ln{n}}$?

The Euler constant is defined as $$\gamma = \lim_{n \to \infty}{\sum_{k=1}^n\frac{1}{k}-\ln{n}}$$ Let $$q = \lim_{n \to \infty}{\sum_{k=2}^n\frac{1}{k\ln{k}}-\ln\ln{n}}$$ I managed to prove that ...
5
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4answers
228 views

How fundamental is Euler's identity, really?

Euler's identity, obviously, states that $e^{i \pi} = -1$, deriving from the fact that $e^{ix} = \cos(x) + i \sin(x)$. The trouble I'm having is that that second equation seems to be more of a ...
7
votes
1answer
67 views

Evaluate $\lim_{R\to\infty}\left(\int_0^R\left|\frac{\sin x}{x}\right|dx-\frac{2}{\pi}\log R\right)$

Is there a closed form of $$\lim_{R\to\infty}\left(\int_0^R\left|\frac{\sin x}{x}\right|dx-\frac{2}{\pi}\log R\right)$$ I am pretty interested whether we can find out a closed form of this limit. We ...
0
votes
1answer
28 views

Euler's Method Solving $y' = f(t,y)$: Programming

I'm trying to compute the approximation solution for $$ y' = -2y + 2 - e^{-4t}: 0\le t\le5, y(0) = 1$$ but the answer that I get for $n = 100$ intervals is $-5.069$ which isn't right. What's the issue ...
1
vote
3answers
36 views

Evaluate $\lim\limits_{n\rightarrow\infty}\left(\frac{n^2 + 1}{n^2 - 2}\right)^{n^2}$

$$\lim\limits_{n\rightarrow\infty}\left(\frac{n^2 + 1}{n^2 - 2}\right)^{n^2}$$ Trying to solve this. At first thought it was $1$, but in wolfram it's $e^3$. Thanks
1
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3answers
26 views

Absolute value of a complex number with a arbitrary basis

I want to calculate the square of the absolute value of a complex number $x^{ia}$, with $x$ and $a$ being real while $i$ is the imaginary number: $$\left|x^{ia}\right|^2=?.$$ I have trouble because ...
0
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1answer
27 views

Limit of exponential function (Eulers number)

I'm struggling with this assignment for a couple of hours, and i dont seem to find any clues :( I have to find the limit of this exponential function: $$\lim_{n \to \infty} (1+\frac{1}{n+2})^{2n+3} ...
0
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0answers
18 views

How to simplify this representation of $\gamma$?

I have a function of two variables, $Z(n,m) = (-1)^m \left( \frac{1}{m-2^n} + \frac{1}{2^n \log (2^n /m)} \right)$, and the infinite sum $\sum_{n=1}^{\infty} \sum_{m=1}^{2^n -1} Z(n,m) = \gamma$, the ...
0
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0answers
37 views

Transform cos to e function

What are the steps in order to transform the cosine function to the exponential function: $$ \left[\cos \left(\frac{k \pi} N\right)\right]^n \approx e ^ {\frac{-n}2 \left(\frac{k \pi} N \right)^2} ...
0
votes
0answers
45 views

Evaluating $\int{e^{-t^{2}}\,dt}$

Let $f(x) = e^{x}$, which, expressed as a Maclaurin series, is equal to: $$\sum_{i = 0}^{\infty}{\frac{x^{i}}{i!}}$$ Therefore, $f(-t^{2})$ gives: $$\sum_{i = 0}^{\infty}{\frac{(-t^{2})^{i}}{i!}} = ...
0
votes
1answer
41 views

Rearrangement of difficult algebraic equations

I have always had difficulty when rearranging large equations of functions to find the roots and also the turning points(using derivatives) and was hoping some people could give me some tips when it ...
1
vote
1answer
60 views

Is the Champernowne constant actually useful

Has the aforementioned constant ever been used in any major proofs? Can it be expressed in terms of $e$, $\pi$, or both? Does it appear in any sort of geometric sense like $\pi$ does? Or is it just a ...
7
votes
1answer
98 views

Integrating:$ \displaystyle \int_0 ^ {\infty}e^{-x^2}\ln(x)dx $

$$ \displaystyle \int_0 ^ {\infty}e^{-x^2}\ln(x)dx = -\frac{1}{4}(\gamma +2\ln(2))\sqrt{\pi} $$ This is a well known integral. But I want to know how to solve it?? Also, please refrain using contour ...
2
votes
1answer
70 views

Is this a valid visualization of Euler's identity as a more generic pattern?

I was reading this nice question about a demonstration of Euler's identity, and tried to visualize how would look the left part of the identity in the complex plane by using the following function: ...
2
votes
1answer
53 views

How can I get $\alpha$ and $\beta$ numerically? (Euler constant)

Let $\gamma_n=\sum_{k=1}^{n}\frac{1}{k}-\ln(n)$ and $\gamma=\lim_{n}\gamma_n$ From the fact $\frac{1}{2(n+1)}\leq\gamma_n-\gamma\leq\frac{1}{2n}$, we have $\gamma_n-\gamma \sim \frac{1}{2n}$ By ...
3
votes
2answers
39 views

How do we bound the function $\frac{-1}{2}\sum_{m=n}^{\infty}\frac{1}{m^{2}}$?

In case the question didn't display in the title correctly: How do we bound the function $\frac{-1}{2}\sum_{m=n}^{\infty}\frac{1}{m^{2}}$? I think a way I can do this is to show that ...
1
vote
2answers
51 views

Error in approximating e via a finite sum

I need some help with a homework problem. I have to find an upper bound for the error in approximating $e$ by the series $$\sum_{k=0}^{n} \frac{1}{k!}$$ I thought about using Taylor's theorem with ...
0
votes
4answers
70 views

Why does the definiton of the Euler's number not violate the rule agaisnt division by zero? [duplicate]

e= appears to be defined as the sum of the series 1/n! as n goes from zero to infinity. But this implies that the first term is 1/0! which appears to violate the rule against division by zero
0
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3answers
85 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
39 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
34 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
170 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
59 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
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1answer
29 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
65 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 ...
1
vote
1answer
354 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
152 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
6answers
178 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 ...
0
votes
1answer
50 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 ...
2
votes
2answers
89 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
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2answers
75 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 ...
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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
128 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 ...
3
votes
2answers
112 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
104 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 ...
3
votes
1answer
66 views

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

I wonder whether $\gamma$ belongs to the ring of periods? UPDATE Well now I know it should not. But $e^{-\gamma}$ should.
2
votes
1answer
66 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
157 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= ...
13
votes
1answer
250 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
52 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
431 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
232 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
88 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 ...
4
votes
1answer
114 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
110 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 ...
42
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} ...