# eulers-constant's questions - Chinese 1answer

226 eulers-constant questions.

### Is there another proof for Euler–Mascheroni Constant?

0 answers, 31 views trigonometry factoring eulers-constant

### 10 Calculating a limit with trignonometeric and quadratic function

$$\lim_{n->\infty}\left(1+{1\over{n^2+\cos n}}\right)^{n^2+n}$$ I vaguely get the idea that since $\cos n$ and $n$ dont really matter compared to $n^2$, this must evaluate to $e$. But not sure how ...

### 5 How to show $\int_{0}^{\infty}{{\gamma+\ln x\over e^x}}\cdot{1-\cos x\over x} \,\mathrm dx={1\over 2}\cdot{\pi-\ln 4\over 4}\cdot{\pi+\ln4\over 4}?$

Consider this integral $(1)$ $$\int_{0}^{\infty}\color{red}{{\gamma+\ln x\over e^x}}\cdot{1-\cos x\over x}\,\mathrm dx={1\over 2}\cdot{\pi-\ln 4\over 4}\cdot{\pi+\ln 4\over 4}\tag1$$ Recall a well-...

### Why is $\int_{0}^{t} e^{nt} \mathrm{\ dt} = \frac{1}{n} \left(e^{nt} - 1\right)$? [solved; notation is also faulty in the first place]

If $e^{nt}$ can also be written as $\left(e^n\right)^t$ or $\left(e^t\right)^n$, $\int_{0}^{t} e^{nt} \mathrm{\ dt}$ can also be written as $\int_{0}^{t} \left(e^{t}\right)^n \mathrm{\ dt}$ which can ...

### -2 Solving Equation with Euler's Number

How to Approximate "$n$" in $$1-e^\frac{-n^2}{2N} = \frac{1}{2}?$$ Textbook Answer:

### Flaw in this reasoning [duplicate]

2 answers, 30 views eulers-constant
$\ e^{i 2\pi} = 1,$ $\ e^{0} = 1$ $\Rightarrow$ $\ e^{i 2\pi} = e^{0}$ $\Rightarrow$ $\ {i 2\pi} = 0$ $\Rightarrow$ $\ i = 0$

### 1 Is it true, that $\sum\limits_{n=1}^{\infty}(e^{\frac{1}{n!e}}-1)=\frac{10}{11+e}$?

2 answers, 76 views summation factorial eulers-constant
If $$\sum\limits_{n=1}^{\infty}(e^{\frac{1}{n!e}}-1)=\frac{10}{11+e}$$ is true, so how can we prove it (if not, how can we came to this approximation)? If I made some mistakes, sorry for my English.

### 8 Intuitively, how would you explain that the logarithm and the number $e$ are related to the function $\frac{1}{x}$ in particular?

6 answers, 99 views logarithms eulers-constant
Why does the integral of the function $\frac{1}{x}$ from 1 to $e$ have to be equal to 1 ? Why does it mathematically make sense? How come a number related to instantaneous, continuous growth has to ...

### 19 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 ...