Power Series and Series

A series is defined in mathematics as adding an infinite number of quantities to a given starting number. Series is used in many areas of mathematics, including the study of finite structures such as combinatorics for forming functions. The series is an important part of calculus and its generalisation and mathematical analysis. The sum of the terms in an arithmetic series can be calculated by multiplying the number of times the average of the last and first terms. Series and finding the sum of series terms are important tasks in mathematics.

For example, an infinite series is the sum of an infinite number of numbers that are related in a specific way and listed in a specific order. In mathematics, as well as in physics, chemistry, biology and engineering, infinite series are useful.

Explanation of Power Series

Infinite series are a subcategory of power series. We will see that in a convergent interval, the sum of the power series is a continuous function with derivatives of all orders. We will also investigate the inverse question, namely, whether a function f = f(x) with derivatives of all orders on an interval can be expressed as a power series. A power series is a function with infinite functional values. As a result, the power series f(x) may be defined for some values of its variable x. Convergence tests will aid in determining the domain of a power series.

There are some important features associated with the generation of power series representations of functions. First, a value of x lying in the domain of the function must be chosen for the expansion point; second, the function must be infinitely differentiable at the chosen point in its domain. In other words, differentiation of the function must never yield a constant function because subsequent derivatives will be zero, and the series will be truncated to a polynomial of finite degree. 

Fourier Series and Taylor Series

Through the partial sums of a power series, the partial sums of the Taylor series approximate a function f(x) in the vicinity of the computation point. If a function must be approximated over a larger interval, terms of very high order are required. The polynomial obtained by truncating the Taylor series must have at least the same number of turning points as the function. This would be tedious for periodical functions with intervals greater than the period.

A Fourier series converges to the function f(x), which it represents at all points where f(x) is continuous and to the mid-point where f(x) has a finite jump discontinuity. These results are sometimes useful when seeking the sum of a simple infinite numerical series, and it is shown how such series can be summed by relating the series to a Fourier series in which the substitution of a particular value of the argument gives rise to the required numerical series.

In principle, at least, the Fourier series would appear to offer the possibility of representing both discontinuous and continuous functions because – whereas for a Taylor series expansion, a function needs to be differentiable – for a Fourier series expansion, it would appear that it only needs to be integrable.

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