Taylor series

If we denote with $y^{(n)}(t)$ the function value at point $t$ of the $n$'the derivative of a function $y$, then the Taylor series of a continuous function $y$ at point $t$ is given by:

$$y(t+h) = y(t) + hy^{(1)}(t) + \frac{1}{2!}h^2y^{(2)}(t) + \frac{1}{3!}h^3y^{(3)}(t) + \cdots + \frac{1}{n!}h^ny^{(n)}(t) + \cdots$$ (1.1)

Some remarks about this expansion:

1. $h$ can be any element of $\mathbb{R}$, such that a function is completely defined by its Taylor expansion at any single point of its domain. (Full knowledge of the function at a single point determines the full function at all points).
2. The successive terms of the Taylor series are decreasing in magnitude in an exponential way.
3. We can truncate a Taylor series to approximate the original function. This divides the Taylor series in two separate series: the numerical scheme and the error series. The error after truncation is mainly dependent on the first term of the error series.