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Questions tagged [eulers-number]

Euler's number is another name for e, the base of the natural logarithms.

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3answers
64 views

Exponential identity ${(x^a)}^b=x^{ab}$

We all know that $e^{\pi i}=\cos \pi + i \sin \pi=-1$, and that ${(x^a)}^b=x^{a \times b}$, also $e^{2 \pi i}=\cos {2\pi} + i \sin {2\pi}=1$. Here's my problem, we have $a \in \mathbb R$, and I ...
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3answers
349 views

Showing $\lim_{n\to \infty} \left( 1+\frac{1}{2n} \right)^n = \sqrt{e}$, intro to analysis

I am in an intro to Analysis class, and I want to show that $$\lim_{n\to \infty} \left( 1+\frac{1}{2n} \right)^n = \sqrt{e}$$ I already have a result that $$\lim_{n\to \infty} \left( 1+\frac{1}{...
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4answers
59 views

What is the derivative of $e^{i\pi}$?

I know that $e^{i\pi}$ =-1 the derivative of $e^{i\pi}$ is the derivative of - 1 which is 0. I guess I'm missing a rule or understood something wrong.
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2answers
78 views

$\sum_{n=0}^{\infty} \frac{\sin(nx)}{3^n}$

I'm currently doing AOPS vol.2 and this problem has stumped me for weeks. Find $$\sum_{n=0}^{\infty} \frac{\sin(nx)}{3^n}$$ Knowing:$$\sin (x)=\frac{1}{3}$$ $$0\leq x\leq \frac{\pi}{2}$$ My Attempt: ...
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0answers
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Intuitive understanding of Euler's Formula

Thanks for reading! In this link... https://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ ...Euler's formula is explained pretty well. I understand that multiplying a ...
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5answers
110 views

Intuitively, why are the two limit definitions of $e^x$ equivalent?

Thanks for reading! Intuitively, why does... $$\lim_{n \rightarrow \infty} \left(1+\frac{1}{n}\right)^{xn}=\lim_{n \rightarrow \infty} \left(1+\frac{x}{n}\right)^{n}=e^x$$ Note, I'm not asking why $...
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2answers
139 views

Is $\left(1+\frac1n\right)^{n+1/2}$ decreasing?

Using the Cauchy-Schwarz Inequality, we have $$ \begin{align} 1 &=\left(\int_n^{n+1}1\,\mathrm{d}x\right)^2\\ &\le\left(\int_n^{n+1}x\,\mathrm{d}x\right)\left(\int_n^{n+1}\frac1x\,\mathrm{d}x\...
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1answer
54 views

Is there a “natural tetration function”?

For the natural exponential function, $$ f(x)=e^x \to f(x)=f'(x). $$ Is there a natural tetration (tetral?) function? $$ f(x)={{^x}b} \to f(x)=f'(x) $$ Is it base $e$?
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1answer
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weird properties of complex exponentials: $e^{i 2 \pi f(x)} = 1$ [duplicate]

$$ \begin{aligned} e^{i 2\pi f(x)} &= (e^{i\ 2\pi})^{f(x)} \\ e^{i 2\pi f(x)} &= (\cos {2\pi} + i\ \sin {2\pi})^{f(x)} \\ e^{i 2\pi f(x)} &= (1 + i 0)^{f(x)} \\ e^{i 2\pi f(x)} &= 1^...
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1answer
65 views

Inequality involving $e$: $\binom{n}{k} < \frac{1}{e}\big(\frac{en}{k}\big)^k$

$\forall n,k\in \mathbb{Z^+}$, prove that $\binom{n}{k} < \frac{1}{e}\big(\frac{en}{k}\big)^k$ where $e$ is the [Euler's number](https://en.wikipedia.org/wiki/E_(mathematical_constant). I tried my ...
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1answer
98 views

Meaning of Maclaurin expansion of $e$

I was wondering if there is an interpretation or specific meaning to the series expansion of $e$. $$ e = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!} + \...
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2answers
56 views

What is the nature of $e^{ix}$, real or complex?

Since, $e^{i\pi}=-1$, right part of the equation ($-1$) is real and left part is seemingly complex as $e^{ix}=\cos(x)+i\sin(x)$ which is a complex number( I am not sure though). I am a freshman, ...
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8answers
6k views

Why is Euler's number $2.71828$ and not anything else? [closed]

Why is Euler's number $\mathtt 2.71828$ and not for example $\mathtt 3.7589$? I know that $e$ is the base of natural logarithms. I know about areas on hyperbola ...
13
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6answers
638 views

Non-Numerical proof of $e<\pi$

This is a "coffee-time-style" problem ( to have a taste of this style, you may like to browse the book https://www.amazon.com/Art-Mathematics-Coffee-Time-Memphis/dp/0521693950) interpreted from an ...
3
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2answers
54 views

$10-e$ interesting decimal expansion property

The number $e$ has several interesting properties but I noticed something that may just be coincidence; a neat coincidence however! Here is the decimal expansion of $e$. $$e=2.718281828459045235360\...
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6answers
198 views

Why is $\lim\limits_{n\to\infty} (1 + \frac{1}{2n})^n = e^{\frac{1}{2}}$

In my textbook it is stated that this is obvious: $\lim\limits_{n\to\infty} (1 + \frac{1}{2n})^n = e^{\frac{1}{2}}$. However I feel stupid for not understanding why? What am I missing?