Taylor series

General Taylor series expansion from calculus: $$ f(x+h) = \sum_{i=0}^\infty \frac{1}{i!}\frac{d^if}{dx^i}(x)h^i\tp $$

Here: expand \( u^{n-1} \) around \( t_n \): $$ \begin{align*} u(t_{n-1}) = u(t-\Delta t) &= \sum_{i=0}^\infty \frac{1}{i!}\frac{d^iu}{dt^i}(t_n)(-\Delta t)^i\\ & = u(t_n) - u'(t_n)\Delta t + {\half}u''(t_n)\Delta t^2 + \Oof{\Delta t^3}, \end{align*} $$

• \( \Oof{\Delta t^3} \): power-series in \( \Delta t \) where the lowest power is \( \Delta t^3 \)
• Small \( \Delta t \): \( \Delta t \gg \Delta t^3 \gg \Delta t^4 \)