Desenvolvimentos em série

$\sin x = x - {{x^3}\over {3!}} + {{x^5}\over {5!}} + \ldots + (-1)^n {{x^{2n+1}}\over {(2n+1)!}} + x^{2n+1} \epsilon(x)$

$\cos x = 1 - {{x^2}\over {2!}} + {{x^4}\over {4!}} + \ldots + (-1)^n {{x^{2n}}\over {(2n)!}} + x^{2n} \epsilon(x)$

$\tan x = x - {{x^3}\over {3}} + {{2x^5}\over {15}} + x^{5} \epsilon(x)$

$\sinh x = x + {{x^3}\over {3!}} + \ldots + {{x^{2n-1}}\over {(2n-1)!}} + x^{2n-1} \epsilon(x)$

$\cosh x = 1 + {{x^2}\over {2!}} + \ldots + {{x^{2n}}\over {(2n)!}} + x^{2n} \epsilon(x)$

$\tanh x = x - {{x^3}\over {3}} + {{2x^5}\over {15}} + x^5 \epsilon(x)$

$\arcsin x = x + {{x^3}\over {6}} + \ldots + {{1.3.5\ldots(2n+1)}\over {2.4.6\ldots(2n)}} {{x^{2n+1}}\over {(2n+1)!}} - x^{2n+1} \epsilon(x)$

$\arctan x = x - {{x^3}\over {3}} + {{x^5}\over {5}} + \ldots + (-1)^n {{x^{2n+1}}\over {(2n+1)!}} + x^{2n+1} \epsilon(x)$

$\sinh^{-1} x = x - {{x^3}\over {6}} + \ldots + (-1)^n {{1.3.5\ldots(2n+1)}\over {2.4.6\ldots(2n)}} {{x^{2n+1}}\over {(2n+1)!}} + x^{2n+1} \epsilon(x)$

$\tanh^{-1} x = x + {{x^3}\over {3}} + {{x^5}\over 5} + \ldots + {{x^{2n+1}}\over {(2n+1)!}} + x^{2n+1} \epsilon (x)$

$e^x = 1 + x + {{x^2}\over {2!}} + {{x^3}\over {3!}} + \ldots + {{x^n}\over {n!}} + x^n \epsilon (x)$

$\log (1+x) = x - {{x^2}\over {2}} + {{x^3}\over {3}} + \ldots + (-1)^{n-1} {{x^n}\over {n}} + x^n \epsilon (x)$

$(1+x)^{\alpha} = 1 + \alpha x + \alpha (\alpha-1) {{x^2}\over {2}} + \ldots + \alpha (\alpha -1) \ldots (\alpha -n+1) {{x^n}\over {n}} + x^n \epsilon (x)$

${1\over {1-x}} = 1 + x + x^2 + \ldots + x^n + x^n \epsilon (x)$

${1\over {1+x}} = 1 - x + x^2 - x^3 + \ldots + (-1)^n x^n + x^n \epsilon (x)$