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## Taylor series

If we denote with the function value at point of the 'the derivative of a function , then the Taylor series of a continuous function at point is given by:

| (1.1)

Some remarks about this expansion:

1. can be any element of , 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.

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