**226 eulers-constant questions.**

Problem
Prove that the sequence $$x_n=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}-\ln n,~~~(n=1,2,\cdots)$$is convergent.
One Proof
This proof is based on the following inequality
$$\frac{1}{n+...

I'm trying to make sense of a solution to a separation of variables question. Namely where it goes from:
$$y = \exp(-\cos x + C)$$
To:
$$y = A\cdot \exp(-\cos x)$$
I understand the constant of ...

I found this question in a book. The answer given is that the roots are $(a+1)^{1/20}\exp\left({\frac{i2k_1\pi}{20}}\right)$ and $(a+1)^{1/20}\exp\left({\frac{i(2k_1+1)\pi}{20}}\right)$. How do I ...

I was reading about how Euler derived his famous identity, $e^{i{\pi}}$. It said that it was discovered when Euler took the Taylor Expansion for $e^x$, and he multiplied the $x$ by $i$, and it gave ...

What's the process, using Euler's Formula, solving $(a+bi)^{a+bi}$ when outlined algebraically in rectangular form?
Edit:
And also solving this in the same form and process, $(d cos(y)+i d sin(y))^{(...

I was looking at the sum $\sum_{i,h=1}^x \frac{1}{i^h}$ on Desmos, and I realized it seemed to converge to the line $y=x$. When I subtracted x from it and increased the bounds, it seemed to be ...

Is there any explicit series, product, integral, continued fraction or other kind of expression for the point at which $\Gamma(x)$ has a minimum in $(0,1)$?
The decimal value can be found here http:/...

It is easily checked that, for each positive integer $k$,$$\frac 1{3k-1}+\frac 1{3k}+\frac 1{3k+1}=\frac 1k+\frac 2{(3k)^3-3k}\tag1$$Set $k=1,2,\ldots,n$ in $(1)$ and add the $n$ equalities to find ...

Given $\dot{x}(t)=Hx(t)+f(t)$ and $x(0)=x_0$, how can I see that the Lagrange's variation
of constants $$x(t)=e^{Ht}x_0+\int_0^te^{(t-s)H}f(s)ds$$ is the right solution to this problem? I'm afraid I ...

I wanted to make one of those cool infinite recursive definitions for myself, and I chose one that I thought looked cool: $x=\sqrt{\sin{x}}=\sqrt{\sin{\sqrt{\sin{\sqrt{\sin{...}}}}}}$ for no other ...

After watching these 2 brilliant YouTube Videos--
1) Euler's formula with introductory group theory
2) But what is the Fourier Transform? A visual introduction.
I am facing some difficulty in ...

$$\lim_{n\to \infty}\int_0^{\pi\over 2}\cos x\cos\left({x\over n } \right) \log(\log\csc (x)) \, dx$$
surprisingly give a result $-0.577156\ldots$, which went we check up it, it is Euler's constant ...

Ok, so I understand $e$ is just $\lim\limits_{n\to \infty}(1 + \frac{1}{n})^n$, and ends up being $2.71828\ldots$
It assumes the annual return you are getting is $100\%$ and is continuously compounded....

I calculated the approximate sums and it seemed to be the case that $x=\gamma$ so far I cannot prove it.

Is there a name for the theorem that $\displaystyle \lim_{n \to \infty} (1+\frac{1}{n})^n < \infty$ ?
Wikipedia has a List of things named after Leonhard Euler which mentions Euler's number but not ...

As the title states, I want to prove
$$\lim_{x \to \infty} \frac{x}{e^x} =0$$
Clearly, L'Hopital's rule easily solves this. However, I'm curious to see if there's another way to prove it, without ...

So I thought you take the inverse function of the whole expression getting:
$x - y = e^{kt}$ and so your final answer would be $x = e^{kt} + y$ but according to the answers in the book $x = ye^{kt}$. ...

I have gotten interested in forecasting using linear/nonlinear regression, particularly using Facebook's Prophet library for R/Python. It makes forecasting on a time-series input pretty ...

We denote the MÃ¶bius function as $\mu(n)$, see its definition from this MathWorld.
On the other hand it is not known if the Euler-Mascheroni constant is irrational.
After I've read a MathOverflow ...

Why does $ e^{i\pi} = -1 $ ?
I know that this form can be used to for instance act on a bloch's sphere (quantum mechanics) using it as $ e^{i\pi/4} $ will do a $ \frac{\pi}{4} $ rotation on the $x-...

I have been asked to calculate how many terms of the series that defines the Euler's constsant $\gamma$ to add at least, to calculate the value of $\gamma$ with error less or equal to $3 \times 10^{-3}...

I think this will be an easy problem for you, but I do not see the solution.
I know that
$$\displaystyle\sum_{k=0}^{\infty} \frac{1}{k!}=e$$
Knowing this, how can I demonstrate this
$$\...

$$
H_n - \ln n = \int_0^1 \frac{1 - x^n}{1 - x} dx - \int_1^n \frac{dy}{y}
$$
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 -...

For example if we have $$ \ln{y}=a\ln{x}$$
If we raise both sides to the power of e:
$$ y = e^a .e^{\ln{x}} = e^ax$$
However by using log rules we get a different solution
i.e. by letting $$a\ln{x}...

$$\lim_{n->\infty}\left(1+{1\over{n^2+\cos n}}\right)^{n^2+n}$$
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 ...

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-...

The definition of $e$ is:
$$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$$
If we use the Binomial Theorem on the function itself:
$$\left(1 + \frac{1}{n}\right)^n = \sum_{k=0}^{n} \binom{...

I looking to evaluate $$\int_{0}^1\ln(\ln(\frac{1}{x})) dx$$
I know the answer is the Euler-Mascheroni constant, $\gamma$ but how do I get that result?
I've tried differentiating under the ...

I am trying to understand the definition and evaluation of $e^x$ and $\log(x)$ and $ln(x)$ and their derivatives, but I can't help but feel that a lot of this stuff is circular. Every time I google ...

I want to prove that$$\lim_{s\to1}\zeta(s)-\log\prod_{n=1}^\infty(1+n^{-s})=\gamma$$
I know that for $|s|>0$, $\log(1+n^{-s})=\sum_{k=1}^\infty (-1)^{k+1}\frac{n^{-sk}}{k}$
So $$\log\prod_{n=1}^\...

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 ...

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

$\ e^{i 2\pi} = 1,$
$\ e^{0} = 1$
$\Rightarrow $
$\ e^{i 2\pi} = e^{0}$
$\Rightarrow $
$\ {i 2\pi} = 0 $
$\Rightarrow $
$\ i = 0 $

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.

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 ...

Introduction
Inspired by the work of Olivier Oloa [1] and the question of Vladimir Reshetnikov in a comment I succeeded in calculating the closed form of the sum
$$s_m = \sum _{k=1}^{\infty } \frac{\...

We know, that
$$e=\lim\limits_{n\to\infty}^{}\left[1+\frac{1}{1}\left(1+\frac{1}{2}\left(1+\frac{1}{3}\left(1+\cdots\frac{1}{n}\right)\right)\cdots\right)\right]$$
If $x_{m}=m(x_{m-1}-1)$ and $x_{0}=e$...

I watched this video by Matt Parker recently:
https://www.youtube.com/watch?v=9tlHQOKMHGA
He calculates $i^i$ and his answer is around ~1/5
well more precise $e^\frac{-\pi}{2}$ which uses
$e^{i\...

If $e$ (Euler's constant) $= (1+\frac{1}{n})^n$ as $n$ approaches infinity, why is $e^x$ not equal to $e$ if x equals any number, real or not real.
Let r = any real number
I think that $\left(1+\...

Title pretty much says it all, but I have a couple of additional questions.
1) Is it even possible?
2) If it is, then where do we start? Like what is the definition of $e$? Is it possible to do this ...

How does one show that $\lim_{x\rightarrow\infty}\left(1+\frac{1}{x}\right)^x$ converges? I have a book that uses this to define $e$ but they don't show that it converges. I suppose we could give it ...

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 ...

Inspired by these three questions, I asked myself whether
$$\sum_{n>0}\Big[~H(e^n)-n-\gamma~\Big]~=~0.278091975548622251874828828459627630\ldots$$
might also possesses a closed form expression, ...

I need help with this limit. I know the result is $e$ and I know I need to somehow modify the exponent into $x$ but am unsure as to how.
$\lim_{(x,y)\to(\infty, 1)}(1 + \frac{1}{x})^{\frac{x^2}{x+y}}$...

According to WolframAlpha, the integral $$\int_0^{\infty} \log(x)^2 e^{-x} \, \mathrm{d}x$$ has closed form $\gamma^2 + \frac{\pi^2}{6}$, where $\gamma$ is the Euler-Mascheroni constant.
The term $\...

I started the proving with the verification of ${\gamma}\lt{1\over\sqrt3}$ inequality (without using calculator). If it is proved then my question is also proved. For this I used the ${e\over {\pi}}\...

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 ...

- sequences-and-series
- calculus
- limits
- integration
- definite-integrals
- logarithms
- real-analysis
- exponential-function
- number-theory
- complex-numbers
- harmonic-numbers
- summation
- approximation
- riemann-zeta
- algebra-precalculus
- gamma-function
- constants
- complex-analysis
- trigonometry
- convergence
- improper-integrals
- closed-form
- irrational-numbers
- pi
- differential-equations

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