Referencia: Lathi Sec. B p54

B.8-3 Sumatorias

\sum_{k=m}^{n} r^k = \frac{r^{n+1}-r^m}{r-1} \texttt{ , } r\neq 1 \sum_{k=0}^{n} k = \frac{n(n+1)}{2} \sum_{k=0}^{n} k^2 = \frac{n(n+1)(2n+1)}{6} \sum_{k=0}^{n} k r^k = \frac{r+[n(r-1)-1]r^{n+1}}{(r-1)^2} \texttt{ , } r\neq 1 \sum_{k=0}^{n} k^2 r^k = \frac{r[(1+r)(1-r^n)-2n(1-r)r^n - n^2(1-r)^2 r^n] }{(1-r)^3} \texttt{ , } r\neq 1

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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)!}