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Lecture 16, 17, 18

# Lecture 16, 17, 18 - MATH 118 18 SEQUENCES 1 Sequences Many...

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MATH 118, LECTURES 16, 17 & 18: SEQUENCES 1 Sequences Many processes in mathematics and the applied sciences result in sequences of numbers. Consider many of the processes of approximation we have carried out in this and the previous semester’s Calculus course. For example, consider the Newton’s method formula x n +1 = x n - f ( x n ) f prime ( x n ) used to estimate the roots of a function (i.e. x * R such that f ( x * ) = 0). What this formula generated was a sequence of estimates for the roots which could have been written as { x 1 , x 2 , x 3 , x 4 , . . . } = { x n } n =1 (1) where the dots represent the fact that the sequence does not terminate at some finite index n —it is an infinite sequence. Similarly, when estimating the area under the curve for functions we could not integrate, we applied various numerical integration techniques to obtain estimates for the area. This could be seen as generating a sequence of numbers; for instance, using the Rectangular Rule we have I n = h n summationdisplay i =1 f ( x i ) = ( b - a ) n n summationdisplay i =1 f parenleftbigg a + ( b - a ) n i parenrightbigg which generates the infinite sequence { I 1 , I 2 , I 3 , I 4 , . . . } = { I n } n =1 . (2) In both of these cases, we were hopeful that the estimates x n and I n became closer and closer approximates to the root of f ( x ) and the actual area under the curve, respectively, as n → ∞ . In terms of the sequences (1) and (2), we were interested in whether or not they converge . If we take x * to be the actual value of the root ( f ( x * ) = 0) and I * to be the actual value 1

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of the area under the curve, we represent that the sequences converge to the limits x * and I * respectively by x n -→ x * , I n -→ I * as n → ∞ or x n n →∞ -→ x * , I n n →∞ -→ I * or lim n →∞ x n = x * , lim n →∞ I n = I * . Whether infinite sequences of numbers converge or not will be an issue of central important to us. (We will discuss the mathematical formulation of convergence a little later on.) There is another important distinction to be drawn from the examples above. The approximate area formula I n depended only on the value of n , so that if we wanted to find, say, I 100 , we could simply plug n = 100 into the formula. We did not need any information whatsoever about the values of I n for n = 1 , . . . , 99—we could simply jump to consideration of the 100 th term is the sequence. Sequences defined this way are said to be defined explicitly . Now consider the formula for the estimates of a root given by x n . In order to evaluate the 100 th number in the sequence, we first need to calculate the 99 th term in the sequence, which depends on the 98 th term, and so on. In fact, we cannot evaluate the estimate at any value of n without having calculated all preceding values in the sequence! Sequences defined in such a manner are said to be defined recursively .
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