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Unformatted text preview: Math 5C – Midterm Review Outline The exam will cover sections 10.110.8, 10.10, 11.111.3. The outline below summarizes the main topics. Note cards and calculators will not be permitted during the exam. (1) Sequences (a) An infinite sequence of real numbers is a function from N to R , written a k , k = 1 , 2 , . . . , or { a k } ∞ k =1 . (b) “lim k →∞ a k = L ” means that the terms of sequence are eventually in every interval containing L . (c) The convergence or divergence of a sequence is not affected by changing a finite number of its terms. (2) Infinite series (a) An infinite series is an expression of the form ∑ ∞ k =1 a k . (b) The partial sums of an infinite series ∑ ∞ k =1 a k is the sequence s n = ∑ n k =1 a k . (c) “ ∑ ∞ k =1 a k converges to a sum S ” means lim n →∞ s n = S . (d) “ ∑ ∞ k =1 a k converges absolutely” means ∑ ∞ k =1  a k  converges. (Absolute conver gence implies convergence.) (e) If ∑ ∞ k =1 a k converges, then lim k →∞ a k = 0. Equivalently, if lim k →∞ a k 6 = 0, then ∑ ∞ k =1 a k diverges. (f) Geometric series (i) A geometric series is a series ∑ ∞ k =1 a k whose terms are in constant ratio, i.e. a k +1 a k = r , for all k . (ii) If r < 1 then the series converges absolutely, and the sum is given by the formula a 1 1 r , i.e. the first term divided by 1 r . (iii) If r ≥ 1, then the geometric series diverges. (g) Convergence tests (i) Ratio test (ii) Integral test (iii) Comparison test (iv) Ratio comparison test (3) Taylor series (a) Given a function f ( x ) with derivatives of every order on an interval  x x  < A , its Taylor polynomial of degree n about x = x is p n ( x ) = n X k =0 f ( k ) ( x ) k ! ( x x ) k , n = 0 , 1 , 2 , . . . . (b) Taylor’s theorem gives a formula for the error f ( x ) p n ( x ) = R n ( x, x ) . (c) If lim n →∞  R n ( x, x )  = 0, for  x x  < A , then f ( x ) = ∞ X k =0 f ( k ) ( x ) k ! ( x x ) k ,  x x  < A. This is called the Taylor series for f ( x ) at x = x ....
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This test prep was uploaded on 04/07/2008 for the course MATH 5c taught by Professor Roche during the Fall '06 term at UCSB.
 Fall '06
 Roche
 Math

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