 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2017/paper-1/15/a/solution

With the first

n

terms counted from the constant term, Taylor's theorem gives

f (x) = k = 0 \sum n - 1 \frac{f ^{(k)} ( a )}{k !} (x - a)^{k} + R_{n} (x), R_{n} (x) = \frac{f ^{(n)} ( ξ )}{n !} (x - a)^{n},

where

ξ

lies between

a

and

x

. Continuous derivatives through order

n

on the joining interval are a sufficient hypothesis for this Taylor theorem with Lagrange remainder. The Taylor remainder measures the error of the finite Taylor polynomial; convergence of the corresponding infinite Taylor series is a separate question.

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