Questions tagged [eulers-number-e]
This tag is for questions relating to Euler Number. Euler's number is another name for e, the base of the natural logarithms.
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The connection between $\pi$, $e$ and $20$ [closed]
It's well documented that $e^{\pi} \approx 20+\pi$. This can be explained using the following series:
$$\sum\limits_{k=1}^{\infty}\frac{8\pi k^{2}-2}{e^{\pi k^{2}}} = 1$$
The series is quickly ...
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When Does ((n^a)-1)/a)) Equal e; A Sophomore's plight
I am a high school student (sophomore) and have come across something I would like explained.
I was watching 3blue1brown for an explanation of calculus, when he used the formula: lim a->0 (d/dx(n^x)...
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Can a parabola be exactly replicated using exponentials?
I came across this interesting artifact of a problem I was helping a friend solve where I used Euler's exponential forms of sine and cosine which later became of the form $e^{ix} + e^{-ix}$ on one ...
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Why does the scalar inside a natural log dissapear when differentiating it? [closed]
For example if I was differentiating $\ln(2x)$ doesn't the chain rule dictate that it should be $2/x$, not $1/x$? Why does the $2$ disappear?
user1329181
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Closed-form expression for the infinite sum in Dobiński's formula
In combinatorial mathematics, I learned about Dobiński's formula for the $n$-th Bell number $B_n$, which states that:
Dobiński's formula gives the $n$-th Bell number $B_n$ (i.e., the number of ...
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How to prove $\lim_{n\to\infty}(1+1/n)^n=\lim_{n\to\infty}1+1/1!+1/2!+...+1/n!$ [closed]
How to prove $\lim_{n\to\infty}(1+1/n)^n=\lim_{n\to\infty}1+1/1!+1/2!+...+1/n!$ rigorously?
I have read many threads regarding the natural constant e, but couldn't find out how to prove this.
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basic complex analysis, confusion about complex exponential and its modulus
This question is from the paper 'A new proof of Spitzer's result on the winding of two dimensional Brownian motion' by R Durrett, 1982.
Let $D_t = A_t + iB_t$ be a complex Brownian motion. Then $\int \...
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Complex Power of a Complex Number Using Euler's Formula
To get the principal answer to a complex number $(a+bi)$ raised to another complex number $(c+di)$ I understand you can get this by first determining $r=\sqrt{a^2+b^2}$ and $\theta = \arctan(b/a)$ and ...
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Why does $f(t)\le g(t)$ imply $\int_a^b f(t)\; dt\le \int_a^b g(t)\; dt$?
I've come across a proof of the classic limit definition of $e = \lim_\limits{n \to \infty}(1 + \frac{1}{n})^n$ that starts with letting $t$ be any (real, I'm assuming) number in the interval $[1, 1 + ...
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How is $\log _{10}(e)=\left[\log _e(10)\right]^{-1}$? [duplicate]
I am watching a logarithm lecture from 3Blue1Brown (great for math dummies like me!)
Here is the video for context:
https://youtu.be/4PDoT7jtxmw?t=1306
The step that I did not follow is when he ...
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Exploring the Limiting Behavior of $(n + 1)^n$ in Base $ n $ as $ n $ Approaches Infinity
Hello Math Stack Exchange Community,
I've been investigating an interesting pattern in the polynomial expansion of $(n + 1)^n$ when expressed in base $ n $, specifically the emergence of a consistent ...
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Why is this for function specific range tending towards $e$
I was messing around with the binomial approximation method as per which $(1+x)^{n} ≈ (1+nx)$ for $x<<1$, so while entering values in the calculator I observed something strange that the value, $...
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A limit with a product and number e [closed]
How can we proof this fact?
$$\lim_{n\to\infty} \sqrt[n] {\displaystyle\prod_{k=1}^{n} e^{- \left (\frac kn \right )^2}}= \frac {1}{\sqrt[3] e} .$$
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Sum of the series $\sum_{k=0}^\infty k^ne^{-k}$ for a positive integer $n$
How can we calculate the sum of the series $\sum_{k=0}^\infty k^ne^{-k}$ for a positive integer $n$? I tried: $$\sum_{k=0}^\infty k^ne^{-k} =(-1)^n \sum_{k=0}^\infty \frac{d^n}{dy^n}\bigg|_{y=1}e^{-ky}...
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$e$ is hidden in Pascal's (binomial) triangle. What is hidden in the trinomial triangle, in the same way?
In Pascal's triangle, denote $S_n=\prod\limits_{k=0}^n\binom{n}{k}$. It can be shown that
$$\lim_{n\to\infty}\frac{S_{n-1}S_{n+1}}{{S_n}^2}=e$$
What is the analogous result for the trinomial triangle?
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