| | For all \(x\):
\[
\begin{aligned}
e^x &= \sum_{k=0}^\infty \frac{x^k}{k!} \\
\cos x &= \sum_{k=0}^\infty (-1)^k \frac{x^{2k}}{(2k)!} \\
\sin x &= \sum_{k=0}^\infty (-1)^k \frac{x^{2k+1}}{(2k+1)!} \\
\cosh x &= \sum_{k=0}^\infty \frac{x^{2k}}{(2k)!} \\
\sinh x &= \sum_{k=0}^\infty \frac{x^{2k+1}}{(2k+1)!} \\
\end{aligned}
\] |
|
For \(\lvert x \rvert < 1\):
\[
\begin{aligned}
\frac{1}{1 - x} &= \sum_{k=0}^\infty x^k \\
\ln(1+x) &= \sum_{k=1}^\infty (-1)^{k+1} \frac{x^k}{k} \\
\arctan(x) &= \sum_{k=0}^\infty (-1)^k \frac{x^{2k+1}}{2k+1} \\
(1 + x)^\alpha &= \sum_{k=0}^\infty {\alpha \choose k} x^k \\
\end{aligned}
\] |
Hyperbolic trigonometric functions
| |
\[
\begin{aligned}
\sinh(x) &= \frac{e^x - e^{-x}}{2} \\
\cosh(x) &= \frac{e^x + e^{-x}}{2} \\
\tanh(x) &= \frac{e^x - e^{-x}}{e^x + e^{-x}} = \frac{\sinh(x)}{\cosh(x)} \\
\end{aligned}
\] |
|
|
# posted by rot13(Unafba Pune) @ 8:49 PM
