 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-1/15/a/solution

For

n \geq 3

n

f (x) = f (a) + f^{'} (a) (x - a) + \frac{f ^{''} ( a )}{2 !} (x - a)^{2} + \dots + \frac{f ^{(n - 1)} ( a )}{( n - 1 )!} (x - a)^{n - 1} + R_{n} (x) .

Under the usual hypothesis that

f

n - 1

times continuously differentiable on the interval joining

a

and

x

, and

n

times differentiable in its interior, the Taylor theorem with Lagrange remainder gives

R_{n} (x) = \frac{f ^{(n)} ( ξ )}{n !} (x - a)^{n}, ξ strictly between a and x .

This is a finite expansion with a controlled remainder; it does not assume an infinite Taylor series converges. For

x = a

the remainder is zero.

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