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Unformatted text preview: 1 Study Guide for Prelim 1 Math 192  Spring 1997 Written by Don Allers; Revised by Sean Carver 1 Improper Integrals Idea: A proper integral is an integral R b a f ( x ) dx, where a and b are both finite and f ( x ) is continuous and finite on [ a,b ]. For proper integrals, the fundamental theorem of calculus tells us that if F is an antiderivative of f (i.e. if F’(x) = f(x)) then R b a f ( x ) dx = F ( b ) F ( a ) . We will define an improper integral as a limit of proper integrals. Comment The integral R b a f ( x ) dx is improper if: (i) a or b is infinite, or (ii) f ( x ) is not finite on all of [ a,b ]. The fundamental theorem of calculus does not apply directly to this class of integrals. Indeed, the definition of a definite integral given in Chapter 4 explicitly assumes that the integral is proper. Without the definitions below, an improper integral would be an expression with no meaning. Solution Method: 1. If a is finite, and f ( x ) is continuous on [ a, ∞ ] we define R ∞ a f ( x ) dx = lim c →∞ R c a f ( x ) dx. 2. If b is finite, and f ( x ) is continuous on [∞ ,b ] we define R b∞ f ( x ) dx = lim c →∞ R b c f ( x ) dx. 3. If a,b are finite, and f ( x ) is continuous on ( a,b ] we define R b a f ( x ) dx = lim c → a + R b c f ( x ) dx. 4. If a,b are finite, and f ( x ) is continuous on [ a,b ) we b define R b a f ( x ) dx = lim c → b R c a f ( x ) dx. 5. If the integral is improper at multiple points on the domain [ a,b ], we divide this domain into subintervals [ a,c 1 ] , [ c 1 ,c 2 ] ,..., [ c n 1 ,c n ] , [ c n ,b ] so that each subinterval contains only one problem point at one of its endpoints, and no problem points in its interior. Then define R b a f ( x ) dx = R c 1 a f ( x ) dx + R c 2 c 1 f ( x ) dx + ... R c n c n 1 f ( x ) dx + R b c n f ( x ) dx, applying the above rules to each piece. Eg: (a) Z 1 x ln xdx, (b) Z ∞ dx x 2 + 1 , (c) Z ∞∞ 2 xdx ( x 2 + 1) 2 , (d) Z 1 1 dx x 2 / 3 . 2 Sequences Idea: Given a sequence of numbers { a n } = a 1 ,a 2 ,a 3 ,... , the question is whether or not these numbers are converging to some limit a . We say that { a n } converges to a , written a n → a , as n → ∞ , if the distance  a n a  between the sequence points and a goes to zero as n increases. Extend the Sequence to a Real Function Idea: Compute the limit of a sequence as the limit of a function f ( x ) as x → ∞ . When? Use this method if the expression defining the sequence a n can be considered as a real function in the variable n whose limit as n → ∞ can be computed easily. Method: To compute the limit of the sequence, take take the limit of its defining expression as you would take the limit of a real function. Remember L’Hopital’s Rule. (The theorem behind this method is the following: If f ( n ) = a n and lim x →∞ f ( x ) = L then lim n →∞ a n = L. 2 2 SEQUENCES Continuous Function Theorem Idea: It would be nice if we could move a limit inside a function, that is, if we could say...
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 Fall '06
 PANTANO
 Calculus, Improper Integrals, Infinite Series, Integrals, Mathematical Series

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