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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1answer
59 views

Implicit Euler method and explicit Euler method

I wanna know what is the difference between explicit Euler's method and implicit Euler's method. And is the local truncation error for both of them is $O(h)$ and the coefficient of the $O(h)$ term is ...
5
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2answers
86 views

Characterizations of $e^x$

I've been thinking about the following problem for a while: (AFAIK) the 'exponential function', $e^x$ can be characterized as the unique solution to the following differential equation with initial ...
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1answer
20 views

Deduction of 2nd order ODE general solutions

When I have some ODE, for example: $$ u''(t) + 5u(t) = 0 $$ I put together a characteristic equation: $$ \lambda ^2 + 5 = 0 $$ Then I compute its roots $r_1$ and $r_2$. And now there are some ...
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2answers
38 views

How can an imaginary equation give a real answer?

I came across this equation, $$ e^{ix} = \cos(x) + i\sin(x) $$ This is the simplified version, the real one is more complex but this part is the one I have a question about. The right side clearly has ...
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3answers
32 views

Is there a non-exponential function whose limit at infinity is a real, irrational number?

$e$, for example, can be calculated through a non-polynomial function $(1+1/x)^x$, but I cant think of an example for a non-exponential function (or rational function) where the limit to infinity ...
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1answer
614 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 ($A_n = 1 + \frac{1}{2} + ... + \frac{1}{n}$) and this ...
2
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0answers
39 views

Representation of e

I was on this wikipedia page https://en.wikipedia.org/wiki/List_of_representations_of_e which has a list of representations of the constant e. I came across this one representation that looked ...
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1answer
313 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 ...
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1answer
18 views

Graph $S$ and its image $f(S)$

$$S=\{z: -1<Re(z)<2\}$$ $$w=f(z)=e^z$$ Graph $S$ and $f(S)= \{w\}=\{f(z)\mid z \in S\}$ My attempt: This is what I think it should look like. Correct me if I'm wrong, but I think the general ...
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1answer
29 views

Approximating end behavior of a function by plugging in infinity

In Algebra 2, I learned to be able to tell if the end behavior of a function has an asymptote, approaches infinity, approaches zero, etc, by plugging in numbers closer and closer to infinity, or by ...
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0answers
22 views

Different Type of Euler's Circuit and Path (Different Looking Graph)

So I have become familiar with the basic graphs that have dots, and lines connection dot to dot. I know that a Euler's circuit is touching every edge of a graph once, and returning to the same point, ...
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2answers
98 views

How to prove this limit about $\gamma=\lim_{N\to \infty }\left(\sum_{n=1}^N\frac{1}{n}-\ln N\right)$

I have no idea how to prove it. $$\lim_{m\rightarrow \infty }\left [ -\frac{1}{2m}+\ln\left ( \frac{e}{m} \right )+\sum_{n=2}^{m}\left ( \frac{1}{n}-\frac{\zeta \left ( 1-n \right )}{m^{n}} \right ) ...
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1answer
502 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} = ...
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2answers
54 views

Where can I find a proof for $\lim\limits_{n \to \infty} \left (n - \Gamma \left( \frac 1n \right) \right) = \gamma$?

I found in a book the following limit: $\lim\limits_{n \to \infty} \left (n - \Gamma \left( \frac 1n \right) \right) = \gamma$. They say that a proof for this is in "Havil, J.: GAMMA, Exploring ...
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2answers
102 views

Proving $\lim_{n\to\infty}\sum_{k=n}^{\infty}\Big(\frac{1}{n\left\lfloor\frac kn\right\rfloor}-\frac1k\Big)=\gamma$

I remember arriving at the following equality: $$\lim_{n\to\infty}\sum_{k=n}^{\infty}\left(\frac{1}{n\left\lfloor\frac kn\right\rfloor}-\frac1k\right)=\gamma$$ where $\gamma$ denotes the ...
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11answers
3k views

What is the fastest/most efficient algorithm for estimating Euler's Constant $\gamma$?

What is the fastest algorithm for estimating Euler's Constant $\gamma \approx0.57721$? Using the definition: $$\lim_{n\to\infty} \sum_{x=1}^{n}\frac{1}{x}-\log n=\gamma$$ I finally get $2$ decimal ...
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1answer
57 views

When can I assume $\operatorname{Log}$ means the natural log?

I have a problem that says: Express $e^{\operatorname{Log}(3+4i)}$ in the form $x+iy$. However, in class we usually only deal with natural log (this problem is from a textbook). How do I know when I ...
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2answers
501 views

Intuitively, why is the Euler-Mascheroni constant near sqrt(1/3)?

Questions that ask for "intuitive" reasons are admittedly subjective, but I suspect some people will find this interesting. Some time ago, I was struck by the coincidence that the Euler-Mascheroni ...
5
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3answers
541 views

Euler-Mascheroni constant expression, further simplification

The Euler-Mascheroni constant gamma is defined as: $$\gamma=\lim\limits_{n \rightarrow \infty}\left(\sum\limits_{m=1}^{n} \frac{1}{m} - \log(n)\right)$$ From this previous question Do these series ...
4
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1answer
72 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 ( ...
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2answers
199 views

Question on Macys formula for Euler-Mascheroni Constant $\gamma$

I think that: $\gamma = \lim_{n\rightarrow\infty} ~~~ 2H_{n} - H_{n(n+1)}~~~~~~$ (where $H_{n}$ is the $n$-th harmonic number) is a closed form of Macys $\gamma$ formula: $\gamma = ...
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1answer
72 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.
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0answers
75 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 ...
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1answer
82 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: ...
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4answers
245 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 ...
3
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2answers
187 views

Need to show that $\lim_{x\to\infty}\left(\sum_{n\le x}^{}\frac{1}{n}-\ln x \right)$ exist and is less than $1$ [duplicate]

Need some help here. I need prove that the following limit exist and is less than $1$ $$\lim_{x\to\infty}\left(\sum_{n\le x}^{}\frac{1}{n}-\ln x\right)$$ I feel a little lost here, this is my first ...
3
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1answer
79 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
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1answer
88 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 ...
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6answers
203 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 ...
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1answer
76 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 ...
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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 ...
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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
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3answers
27 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 ...
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1answer
35 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} ...
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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} ...
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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 ...
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0answers
46 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!}} = ...
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1answer
50 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 ...
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1answer
62 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 ...
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1answer
107 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 ...
13
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1answer
255 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 ...
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1answer
58 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 ...
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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 ...
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2answers
53 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 ...
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4answers
73 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
87 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 ...
43
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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} ...
1
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1answer
41 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
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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
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1answer
198 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 ...