Paper 3 Practice

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This questions asks you to investigate the recursive nature of the derivatives of 1−x1 and how this impacts the Maclaurin series expansion.

Consider the function f(x)=1−x1, x=1.

Let f(n) be the nth derivative of f.

Using mathematical induction, prove that (1−x)f(n)(x)−nf(n−1)(x)=0 for all n∈N.
[6]
Hence solve the differential equation dxdy=1−x3y.
[3]
Using (d), show that f(n)(0)=n! for all n∈N.
[2]
Hence find the Maclaurin series expansion for f(x).
[3]
Use the Maclaurin series expansion to find the value of f(21), and comment on the result.
[3]