| | For all x:
ex=∞∑k=0xkk!cosx=∞∑k=0(−1)kx2k(2k)!sinx=∞∑k=0(−1)kx2k+1(2k+1)!coshx=∞∑k=0x2k(2k)!sinhx=∞∑k=0x2k+1(2k+1)! |
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For |x|<1:
\begin{aligned}
\frac{1}{1 - x} &= \sum_{k=0}^\infty x^k \\
\ln(1+x) &= \sum_{k=1}^\infty (-1)^{k+1} \frac{x^k}{k} \\
\arctan(x) &= \sum_{k=0}^\infty (-1)^k \frac{x^{2k+1}}{2k+1} \\
(1 + x)^\alpha &= \sum_{k=0}^\infty {\alpha \choose k} x^k \\
\end{aligned}
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Hyperbolic trigonometric functions
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\begin{aligned}
\sinh(x) &= \frac{e^x - e^{-x}}{2} \\
\cosh(x) &= \frac{e^x + e^{-x}}{2} \\
\tanh(x) &= \frac{e^x - e^{-x}}{e^x + e^{-x}} = \frac{\sinh(x)}{\cosh(x)} \\
\end{aligned}
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# posted by rot13(Unafba Pune) @ 8:49 PM
