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Physics 116A Winter 2011 Pointwise and Uniform Convergence 1. Mathematical Definitions



A power series, f(x) = anx n, n=0 is an example of a sum over a series of functions f(x) = gn(x) , (1) n=0 where gn(x) = anx n. It is useful to consider the more general case. Let us consider a sum of the form given in eq. (1) and ask whether the sum is convergent. If we consider each x separately, then we can determine whether the sum converges at the point x. Suppose that the sum is determined to converge for all points x ∈ A, where A is some interval on the real axis. Typical intervals are: the open interval a < x < b, which we will denote by (a, b) and the closed interval a ≤ x ≤ b, which we will denote by [a, b]. Of course, we could also consider half-open, half-closed intervals, such as a < x ≤ b, denoted by (a, b] and a ≤ x < b, denoted by [a, b). In this notation, the parenthesis indicates that the endpoint is not included in the interval, whereas the square bracket indicates that the endpoint is included in the interval. If f(x) converges for all x ∈ A, we say that the sum given by eq. (1) is pointwise convergent over the interval x ∈ A. In this case, A is called the interval of convergence. A classic example is the infinite geometric series, 1 1 − x = x n, |x | < 1. (2) n=0 The above sum converges pointwise to the function (1 − x) −1 inside the open interval −1 < x < 1. Using the standard procedures, it is easy to see that the sum diverges for all |x | ≥ 1. Let us return to the general case of f(x) = gn(x) , x ∈ A, n=0 where A is the interval of convergence. Suppose that the gn(x) are continuous functions. Does this imply that f(x) is continuous? The great mathematician 1 Augustin Louis Cauchy got the answer wrong. In 1821, he claimed to prove that all infinite sums of continuous functions are continuous. It took over 30 years before the error was properly corrected. It is simple to provide a counterexample to Cauchy’s claim. Consider the series: ∞ ∑ x f(x) = gn(x) = 2 (1 + nx2)[1 + (n − 1)x2. (3) n=1 n=1 Although this series looks complicated, we can simplify it using partial fractions. The following is an algebraic identity: gn(x) ≡ x 2 (1 + nx 2)[1 + (n − 1)x 2]

Year: 2014
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