Sequences and Series of Functions

The fact that polynomial functions are so ubiquitous in both pure and applied analysis can be attributed to any number of reasons. They are continuous, infinitely differentiable, and defined on all of R.They are easy to evaluate and easy to manipulate, both from the points of view of algebra (adding, multiplying, factoring) and calculus (integrating, differentiating). It should be no surprise, then, that even in the earliest stages of the development of calculus, mathematicians experimented with the idea of extending the notion of polynomials to functions that are essentially polynomials of infinite degree. Such objects are called power series, and are formally denoted by $$\sum\limits_{n = 0}^\infty {{a_n}} {x^n} = {a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^4} + \ldots $$.