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$\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x- \frac{x^3}{3!} + \frac{x^5}{5!}-\frac{x^7}{7!}+ \cdots, \m |x|<\infty$
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$\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1- \frac{x^2}{2!} + \frac{x^4}{4!}-\frac{x^6}{6!}+ \cdots,\m |x|<\infty $
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$e^x = \sum_{n=0}^{\infty} \frac{ x^{n}}{n!} = 1+x+ \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots, \m |x|<\infty $
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$\ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^{n}}{n} = x- \frac{x^2}{2 } + \frac{x^3}{3 } -\frac{x^4}{4 } + \cdots, \m -1< x \le 1 $
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$(1+x)^p = \sum_{n=0}^{\infty} \pmatrix{p\cr n} x^{n} = 1+p x+ \frac{p(p-1)}{2! }x^2 + \frac{p(p-1)(p-2)}{3! } x^3 + \cdots, \m |x|<1 $
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Don't forget the formula from Lecture 1:
$\mm \frac1{1-x} = \sum_{n=0}^{\infty} x^{n} = 1 + x + x^2 + x^3 + x^4 + \cdots, \m |x|<1 $