Showing posts from January, 2021


Computer Science Style Guide for Python Code Mathmatics AoPS: Theorems The Mathematics Genealogy Project Understanding Quaternions

Maclaurin Series and Euler's Formula

Maclaurin Series: $\displaystyle f(x)=a_0+a_1x+a_2x^2+\cdots +a_nx^n+\cdots\ \ \ and \ \ a_n=\frac{f^n(0)}{n!}$ $\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\cdots +{\frac {x^{n}}{n!}}+\cdots \quad \forall x$ $\displaystyle \sin x =\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-\cdots\forall x $ $\displaystyle \cos x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots \forall x$ Euler's Formula: $\displaystyle \boxed{e^{ix}=\cos x +i\sin x}$ $\displaystyle =1+ix-\frac{x^2}{2!}-i\frac{x^3}{3!}+\frac{x^4}{4!}+i\frac{x^5}{5!}+\cdots=\Big(1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots\Big)+i\Big(x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-\cdots\Big)$



The Benefits of Globalization

A Grade 10 position paper against the following quote. "Globalization, as defined by rich people like us, is a very nice are talking about the internet, you are talking about cell phones, you are talking about computers.  This doesn't affect two-thirds of the people in the world."  - Jimmy Carter The Benefits of Globalization Ever since the Industrial Revolution, globalization has made goods cheaper and more accessible to the general population than ever before.  However, one such critic in Jimmy Carter states that "rich people," in which he is referencing citizens of developed countries such as the United States, are the main benefactors from globalization, because they have access to resources such as "the Internet,...cell phones,...[and] computers."  He also states that globalization "doesn't affect two-thirds of the people of the world," referring to citizens of developing countries.  It is also worth noting that Carter wa