Expansiones de Series

Referencia: Leon W Couch Apéndice p658

Series Finitas

$\sum_{n=1}^{N} n = \frac{N(N+1)}{2}$ $\sum_{n=1}^{N} n^2 = \frac{N(N+1)(2N+1)}{6}$ $\sum_{n=1}^{N} n^3 = \frac{N^2(N+1)^2}{4}$ $\sum_{n=0}^{N} a^n = \frac{a^{N+1}-1}{a-1}$ $\sum_{n=0}^{N} \frac{N!}{n!(N-n)!}x^n y^{N-n} = (x+y)^N$ $\sum_{n=0}^{N} e^{j(\theta+n\phi)} = \frac{sen \left[(N+1) \frac{\phi}{2}\right] }{sen \left( \frac{\phi}{2} \right)} e^{j [ \theta + \left( N \frac{\phi}{2} \right) ]}$ $$

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$\sum_{n=0}^{N} {N \choose k} a^{N-k}b^{k} = (a+b)^N,$ $donde: {N \choose k} = \frac{N!}{(N-k)!k!}$

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Series Infinitas

Serie de Taylor

$f(x) = \sum_{n=0}^{\infty} \left( \frac{f^{(n)}(a)}{n!} \right) (x-a)^n$

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Serie de Fourier

$f(x) = \sum_{n=-\infty}^{\infty} c_n e^{jn\omega_0 x}$ $a\leq x \leq (a+T)$ $donde: c_n = \frac{1}{T} \int_{a}^{a+T} f(x) e^{-jn\omega_0 x} dx$ $\omega_o = \frac{2\pi}{T}$

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otras series

$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$ $sen(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$ $cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$