Maclaurin Series

$f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + ...$

$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...$

$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - ...$

$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - ...$

$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - ...$ ($|x| \leq 1$)

$(1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + ...$ ($|x| < 1$)

Find the Maclaurin series for $f(x) = e^{2x}$ up to $x^3$.

$f(0) = 1$, $f'(0) = 2$, $f''(0) = 4$, $f'''(0) = 8$.

$e^{2x} \approx 1 + 2x + 2x^2 + \frac{4x^3}{3}$.

Or substitute $2x$ into $e^x$ series: $1 + 2x + \frac{(2x)^2}{2} + \frac{(2x)^3}{6} = 1 + 2x + 2x^2 + \frac{4x^3}{3}$.