About \(x = 0\):
| |
\[
\begin{aligned}
\frac{1}{1-x} &= 1 + x + x^2 + \cdots \\
e^x &= 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \\
\ln(1+x) &= x - \frac{x^2}{2} + \frac{x^3}{3} + \cdots \\
\cos x &= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots \\
\sin x &= x - \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots \\
\arctan x &= x - \frac{x^3}{3} + \frac{x^5}{5} + \cdots \\
\end{aligned}
\] |
About \(x = a\):
| |
\[
\begin{aligned}
f(x) = f(a) + f'(a) (x-a) + f''(a) \frac{(x-a)^2}{2!} + f'''(a) \frac{(x-a)^3}{3!} + \cdots
\end{aligned}
\] |
Let \(h = x - a\):
| |
\[
\begin{aligned}
f(a + h) = f(a) + f'(a) h + f''(a) \frac{h^2}{2!} + f'''(a) \frac{h^3}{3!} + \cdots
\end{aligned}
\] |
# posted by rot13(Unafba Pune) @ 11:39 AM
