section_path: Series of functions/Power series
section_name: Power series
paragraph_text: Power series
  
  A power series is a series of the form
  
  ∑ n = 0 ∞ a n (x − c) n. {\displaystyle \sum _{n=0}^{\infty }a_{n}(x-c)^{n}.}
  
  The Taylor series at a point c of a function is a power series that, in many cases, converges to the function in a neighborhood of c. For example, the series
  
  ∑ n = 0 ∞ x n n ! {\displaystyle \sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}}
  
  is the Taylor series of e x {\displaystyle e^{x}} at the origin and converges to it for every x.
  
  Unless it converges only at x = c, such a series converges on a certain open disc of convergence centered at the point c in the complex plane, and may also converge at some of the points of the boundary of the disc. The radius of this disc is known as the radius of convergence, and can in principle be determined from the asymptotics of the coefficients a. The convergence is uniform on closed and bounded (that is, compact) subsets of the interior of the disc of convergence: to wit, it is uniformly convergent on compact sets.
  
  Historically, mathematicians such as Leonhard Euler operated liberally with infinite series, even if they were not convergent. When calculus was put on a sound and correct foundation in the nineteenth century, rigorous proofs of the convergence of series were always required.