In the Taylor series (4.85) of Prob. 4.1, let the point x = a be called xn (that is, xn ‰¡ a) and let s ‰¡ x - a = x - xn, so x = xn + s.
(a) Use this notation to write (4.85) as f(xn + s) equal to a power series in s and evaluate all terms through s5.
(b) In the result of part (a), change s to - s to find a series for f(xn - s). Then add the two series and neglect terms in s6 and higher powers of s to show that

where the notation of (4.65) with ψ replaced by f was used and then f was replaced by ψ.
(c) Replace f in (4.87) by ψ, multiply the resulting equation by s2, neglect the s6 term, solve for ψn(iv) s4, and use ψ" = Gψ [Eq. (4.66)] to show that

Substitute (4.89) and ψn = Gnψn into (4.88) and solve for ψn+1 to show that Eq. (4.67) holds.

Transcribed Image Text:

n+1 (4.87) (4.88) (4.89)