 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2001/paper-1/9/solution

For an

N + 1

times continuously differentiable function, Taylor's theorem gives

f (x) = n = 0 \sum N \frac{f ^{(n)} ( x _{0} )}{n !} (x - x_{0})^{n} + \frac{f ^{(N + 1)} ( ξ )}{( N + 1 )!} (x - x_{0})^{N + 1},

where

ξ

lies between

x_{0}

and

x

. An infinite Taylor series represents the function when its remainder tends to zero; this holds locally for an analytic function, rather than for every smooth function automatically.

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