Step By Step Calculus » 17.0 - History and Applications

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History and Applications
In the 14th century, Madhava of India first conceived the idea of an infinite series expansion of a function. He discovered a number of infinite series, including the Taylor series of the trigonometric functions of sine, cosine, tangent and arctangent. In the 17th century, James Gregory also discovered several Maclaurin series. In 1715, English mathematician Brook Taylor provided a general method for constructing the Taylor series for all functions for which they exist.
We have mentioned earlier that the functions have uses in defining scientific, engineering, or business processes. However, most functions are neither representable nor computable by machines (e.g. computers, calculators) in the way they are defined. This is the place where series becomes useful since many functions can be represented or approximated by certain series, and these series representations of functions help one to evaluate the functions in computers.
It is also common to find series solutions to differential equations that occur throughout Engineering, Science and Business. This series solutions take the form of power series \displaystyle y(x)=\sum_{i=0}^{\infty} a_i x^i\displaystyle y(x)=\sum_{i=0}^{\infty} a_i x^i where one needs to find the coefficients a_ia_i for i=0,1,2,\cdotsi=0,1,2,\cdots.