 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-1/15/a/solution

n

terms of Taylor's theorem, with the Taylor remainder in mean-value form, are

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

for

x \neq = a

. Sufficient hypotheses are continuous derivatives through order

n - 1

on the interval between the endpoints and an

n

th derivative in its interior. At

x = a

the remainder is zero. The finite Taylor polynomial ends at degree

n - 1

; this identity does not require convergence of an infinite Taylor series.

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