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Notes
Popular Definitions
\(e \approx 2.71828 \)
\(e= \displaystyle \lim_{n\to \infty} \left(1+\frac{1}{n}\right)^n \)
\(e= \displaystyle\sum_{n=0}^{\infty}\frac{1}{n!}\)
\(e= \displaystyle \lim_{x\to 0^+} x^{\frac{1}{\ln{x}}} \)
\(\ln(x) = \log_e (x) \)
Limits & Definitions
\(e = \displaystyle \lim_{x\to 0}\left(1+x\right)^{1/x}\)
\(1 = \displaystyle \lim_{x\to 0}\frac{\ln(1+x)}{x}\)
\(e = \displaystyle \lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}}\)
Calculus Magic
\(\frac{d}{dx}\left(e^x\right)=e^x\)
\(\int e^x\,dx = e^x + C\)
\(\int_1^e \frac{1}{x}\,dx = 1\)
\(\ln(x)=\displaystyle\int_1^x \frac{1}{t}\,dt\)
\(\lim_{h\to 0}\frac{e^h-1}{h}=1\)
Series & Expansions
\(e^x=\displaystyle\sum_{n=0}^{\infty}\frac{x^n}{n!}\)
\(\ln(1+x)=\displaystyle\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^n}{n}\quad(|x|\le1,\ x\ne -1)\)
Differential Equations
\(y’=y \quad\Rightarrow\quad y=Ce^x\)
\(\frac{dP}{dt}=kP \quad\Rightarrow\quad P(t)=P_0e^{kt}\)
Probability & Randomness
\(P(X=k)=\displaystyle\frac{e^{-\lambda}\lambda^k}{k!}\)
\(\lim_{n\to\infty}\frac{!n}{n!}=\frac{1}{e}\)
\(\lim_{n\to\infty}\left(1-\frac{1}{n}\right)^n=\frac{1}{e}\)
Complex Analysis
\(e^{ix}=\cos x+i\sin x\)
\(e^{i\pi}+1=0\)
Weird but Awesome
\(\lim_{n\to\infty}n\left(\sqrt[n]{n!}-\frac{n}{e}\right)=\frac{1}{2}\)
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