{"id":4341,"date":"2013-12-15T06:05:38","date_gmt":"2013-12-14T21:05:38","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=4341"},"modified":"2014-08-01T19:19:41","modified_gmt":"2014-08-01T10:19:41","slug":"uniform-convergence","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=4341","title":{"rendered":"Uniform convergence"},"content":{"rendered":"
\n

The ideas of previous article can be extended to the case where the un<\/sub> are functions of x denoted by un<\/sub>(x). In such case the sequences or series will converge of diverge according to the particular value of x. The set of values of x for which a sequence or series converges is called the region of convergence, denoted \\cal R. <\/p>\n

The series u1<\/sub>(x) + u2<\/sub>(x) + … converges to the sum S(x) in a region \\cal R if given ε > 0 there exists a number N, which in general depends on both ε and x, such that |S(x) – Sn<\/sub>(x)| < ε whenever n > N where Sn<\/sub>(x) = u1<\/sub>(x) + … + un<\/sub>(x). If you can find N depending only on ε and not on x, the series converges uniformly to S(x) in \\cal R. Uniformly convergent series have many important advantages as indicated in the following theorems. <\/p>\n

    \n
  1. If un<\/sub>(x), n = 1, 2, 3, … are continuous in a ≤ x ≤ b and ∑ un<\/sub>(x) is uniformly convergent to S(x) in a ≤ x ≤ b, then S(x) is continuous in a ≤ x ≤ b.<\/li>\n
  2. If ∑u<\/sub>(x) converges uniformly to S(x) in a ≤ x ≤ b and un<\/sub>(x), n = 1, 2, 3, … are integrable in a ≤ x ≤ b, then
    \n\\displaystyle \\int^{b}_{a}S(x)dx = \\int^{b}_{a}(u_1(x) + u_2(x) + \\cdots)dx = \\int^{b}_{a}u_1(x)dx + \\int^{b}_{a}u_2(x)dx + \\cdots<\/li>\n
  3. If un<\/sub>(x), n = 1, 2, 3, … are continuous and have continuous derivatives in a ≤ x ≤ b and if ∑un<\/sub>(x) converges to S(x) while ∑u’n<\/sub>(x) is uniformly convergent in a ≤ x ≤ b, then
    \n\\displaystyle S'(x) = \\frac{d}{dx}(u_1(x) + u_2(x) + \\cdots) = u'_1(x) + u'_2(x) + \\cdots<\/li>\n
  4. If there is aset of positive constants Mn<\/sub>, n = 1, 2, 3, … such that |un<\/sub>| ≤ Mn<\/sub> in \\cal R and ∑Mn<\/sub> converges, then ∑un<\/sub>(x) is uniformly convergent [and also absolutely convergent] in \\cal R. <\/li>\n<\/ol>\n

    An important test for uniform convergence, often called the Weierstrass M test, is given by the above. <\/p>\n<\/blockquote>\n