# In the Taylor series (4.85) of Prob. 4.1, let the point x = a be called xn

## Question:

(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 Ïˆ replaced by f was used and then f was replaced by Ïˆ.

(c) Replace f in (4.87) by Ïˆ, multiply the resulting equation by s2, neglect the s6 term, solve for Ïˆn(iv) s4, and use Ïˆ" = GÏˆ [Eq. (4.66)] to show that

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

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