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Definition of Power Series
9.7 POWER SERIES So far we have studied series of constants, One can also form a series of functions Such a series will converge for some values of x and diverge for others. The sum of the series is a new function which is defined at each point x_{0} where the series converges. We shall concentrate on a particular kind of series of functions called a power series. Its importance will be evident in the next section where we show that many familiar functions are sums of power series. DEFINITION A power series in x is a series of functions of the form The nth finite partial sum of a power series is just a polynomial of degree n, The infinite partial sums are polynomials of infinite degree, At x = 0 every power series converges absolutely, (In a power series we use the convention a_{0}x^{0} = a_{0}.) If a power series converges absolutely at x = u, it also converges absolutely at x = u, because the absolute value series and are the same. Intuitively, the smaller the absolute value x, the more likely the power series is to converge at x. This intuition is borne out in the following theorem.


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