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: $\displaystyle \boxed{e^{ix}=\cos x +i\sin x}$
$\displaystyle =1+ix-\frac{x^2}{2!}-i\frac{x^3}{3!}+\frac{x^4}{4!}+i\frac{x^5}{5!}+\cdots=\Big(1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots\Big)+i\Big(x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-\cdots\Big)$

Comments

  1. Titanium Cost of building a mobile casino in a virtual reality
    The company has unveiled tittanium a new winnerwell titanium stove design that uses 3D models from the world's top manufacturers – how to get titanium white octane like Spinit, Vision and Nucleus – to pure titanium earrings create an immersive, titanium dioxide formula

    ReplyDelete
  2. A on line casino no-deposit bonus is a great way|a good way|an effective way} of incomes free cash with out having to make a deposit. You can check out a new new} on line casino or see how well the software program works in your smartphone. You may additionally need to 1xbet korea verify withdrawal processing times.

    ReplyDelete
  3. They’re always including extra video games to their betting library, so you’ll discover something new to explore each time you log in. The progressive jackpot can happen on considered one of 50 pay traces with ninety four.75% RTP. In The United States, a web-based casino can be 우리카지노 registered in Delaware and Nevada states.

    ReplyDelete

Post a Comment

Popular posts from this blog

Harmonic Series

High School Physics - Kinematics