 By Richard Durrett

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Extra resources for Brownian Motion and Martingales in Analysis (The Wadsworth Mathematics Series)

Sample text

6. We note first that the exponential function Æx has derivatives of all orders for every real number x. The Maclaurin polynomial is n 6 k 0 1 k 1x xk x2 2 … xn n from which we have n Æ Æ1  6 k 0 1k k 11 1 2 … 1 n . 000005. 1 with f +x/ being the interval [0,1]. 17) +n  1/ where M is an upper bound on the value of f +n1/ +x/ Æx for x in the interval #0, 1'. However, Æx is an increasing function, so its maximum value on the interval #0, 1' occurs at x 1; that is, Æx  Æ on this interval.

K 1 The number sn is called the partial sum of the series and the sequence sn  n 1 is called the sequence of partial sums. As n increases, the partial sum sn u1  u2  u3  …  un includes more and more terms of the series. Mathematics for Engineers 38 Thus if sn tends toward a limit as n  , it is reasonable to view this limit as the sum of all the terms in the series. 6. 6. Convergent Infinite Sum Let sn  be the sequence of partial sums of the series u1  u2  u3  …  uk  … If the sequence sn  converges to a limit S, then the series is said to converge to S, and S is called the sum of the series.

Thus, limn +1/n1 1 n 0 which is also derived via lim n +1/n1 n 0 so the sequence converges to 0. Ô If the general term of a sequence is f +n/, and if we replace n by x, where x can vary over the entire interval #1, /, then the values of f +n/ can be viewed as sample values of f +x/ taken at the positive integers. 10). 11). 10. Replacement of a sequence  n 20 ! 2 n1 n 1 x by a function f +x/ 2 x1 . 11. Replacement of a function f +x/ x 2 x1 by a sequence  n 10 ! 2 n1 n 1 . 10. L'Hopital's Rule Find the limit of the sequence n s Æn  n 1.