By Richard Durrett
This booklet might be of curiosity to scholars of arithmetic.
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Additional resources for Brownian Motion and Martingales in Analysis (Wadsworth & Brooks Cole Mathematics Series)
56) that x form of the expansion is wrong. 58) v>0 and determine v in the course of analysis. 59) balance each other, 2 v must be equal to 1 or v - \, and x , = ±y/Ji. 58) that the second and third roots are given by x = 1±e 1 / 2 VJi + - •• This example illustrates the fact that difficulties arise whenever the assumed form of the expansion is not correct. But once the form is corrected, a consistent solution is obtained. This is typical of perturbation problems. 60) 42 ALGEBRAIC EQUATIONS in which the small parameter multiplies the highest power of x.
72) Extracting the dominant terms, we have e -nv) 0 n _ e -mVym Hence. VJUAUUNS y~ n = I = m e 45 2 i > w where r = 1 , 2 , 3 , . . , ( « - m ) . Hence, j = c o , c o , . . r ) (2-75) where k = n - m. We discard the root j> = 0 because it corresponds to the first s roots. 72) as ny - x e n 1 0 = my - x e' v m 1 +a / 0 m . y l m - + •• • 1 Hence, equating the coefficients of e on both sides yields v ny ~ x n l =my - x +a - y ~ m l 0 Q m l m x Hence, FT _ m-\ a ~y~ 0 n n \ y r^ 1 — -my" ' x a _ m-l m-l ny ' 1 1 N RT _ ,~ a -m"n-m n m 7 ^x ' K } Therefore, the last n - m roots are given by x = ( ± e + JlLLL .
CO + X X and then \R (CJ)\ N 1 =— f ° ° x * e~ — N Jf. 68) CO Hence, the error committed in truncating the series after N terms is numerically less than the first neglected term, namely the (N + l)th term. Moreover, as co -*• °° with N fixed, R N -* 0. 64) diverges, for a fixed N the first TV terms in the series can represent /(co) with an error that can be made arbitrarily small by taking co sufficiently large. 71) can be rewritten as N ' / M - Z 1 a„ ( 1\ T ^ b r ) 0-72) We should note that the utility of an asymptotic series lies in the fact that the error committed in truncating the series is by definition the order of the first neglected term, and hence, it tends rapidly to zero as co °°.