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MATH 140B HW 5 SOLUTIONS Problem1(WR Ch 7 #8) If I(x) ˘ 0 (x •0),1 (x ¨0),if {xn} is a sequence of distinct points of (a,b), and ifP jcnj converges, prove that the series f (x) ˘ X1 n˘1 cnI(x¡xn) (a •x •b) converges uniformly, and that f is continuous for every x 6˘xn Solution Let fk(x) ˘ Xk n˘1 cnI(x¡xn) By the Weierstrass Mtest (Theorem 710) with Mn ˘jcnj, {fk(x. Where b is a positive real number, and the argument x occurs as an exponent For real numbers c and d, a function of the form () = is also an exponential function, since it can be rewritten as = () As functions of a real variable, exponential functions are uniquely characterized by the fact that the growth rate of such a function (that is, its derivative) is directly proportional to the. Since f(−x) = e− (− x) 2 2 = e− 2 = f(x) and lim x→±∞ e− (−x)2 2 = 0, the graph is symmetry wrt the yaxis, and the xaxis is a horizontal.
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EC02 Spring 06 HW5 Solutions February 21, 06 6 Problem 343 • X is an Erlang (n,λ) random variable with parameter λ = 1/3 and expected value EX =. De nition 4 A function f X!Y is sequentially continuous at a2Xif x n!ain Ximplies that f(x n) !f(a) in Y Theorem 5 A function f X!Y is continuous at aif and only if it is sequentially continuous at a. May 06, 21 · ID3 TSSE Lavfÿó€Info Q2BHù !$&(,0368;=@CEGJMOQTWY\^adfhknpsvxz}€‚‡‰ ‘”—™œž¡£¦¨«®°²¶¸º½ÀÂÅÇÉÌÏÑÔ.
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