 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/discrete-mathematics-continuous-mathematics-and-probability-resit/4/a/solution

One standard finite form of Taylor's theorem is the following. Suppose

f

has continuous derivatives through order

m + 1

on an interval containing

a

and

x

. Then, if

x \neq = a

, there is some

ξ

strictly between them such that

f (x) = k = 0 \sum m \frac{f ^{(k)} ( a )}{k !} (x - a)^{k} + \frac{f ^{(m + 1)} ( ξ )}{( m + 1 )!} (x - a)^{m + 1} .

The final term is the Lagrange remainder. At

x = a

the identity holds directly with zero remainder. The finite expansion requires no assumption that an infinite Taylor series converges to

f

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