Скачать или смотреть Euler's formula power series derivation. Application to e^(i*pi)=-1 derivation.

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In this video, we derive Euler's formula with power series expansions. After deriving the Maclaurin series (power series centered at zero) for e^x, sin(x) and cos(x), we can't help but think there must be some way to put together sin(x) and cos(x) to build e^(x), since the sine function has all the odd terms and the cosine function has all the even terms. In this Euler's formula power series derivation, we try it by evaluating the series for e^(ix), which turns out to give us Euler's formula when we rearrange the terms. So, we arrive at e^(ix)=cos(x)+isin(x) which is called Euler's formula which gives us the tools to express complex numbers in polar form. Finally we apply Euler's formula to obtain the formula on the most popular math t-shirt of all time: e^(i*pi)=-1.