MT-Review - Math 5C Midterm Review Outline The exam will...

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Math 5C – Midterm Review Outline The exam will cover sections 10.1-10.8, 10.10, 11.1-11.3. 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 = 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 0 | < A , its Taylor polynomial of degree n about x = x 0 is p n ( x ) = n k =0 f ( k ) ( x 0 ) k ! ( x - x 0 ) k , n = 0 , 1 , 2 , . . . . (b) Taylor’s theorem gives a formula for the error f ( x ) - p n ( x ) = R n ( x, x 0 ) . (c) If lim n →∞ | R n ( x, x 0 ) | = 0, for | x - x 0 | < A , then f ( x ) = k =0 f ( k ) ( x 0 ) k ! ( x - x 0 ) k , | x - x 0 | < A.
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This is called the Taylor series for f ( x ) at x = x 0 . (4) Power series (a) A power series is an infinite series of the form k =0 a k ( x - x 0 ) k .
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