TMUA Practice · Paper 2 · Question 6

The sequence of functions $f_{1} (x), f_{2} (x), f_{3} (x), \dots$ is defined as follows:

f_{1} (x) = x^{10}

f_{n + 1} (x) = x f_{n}^{'} (x) for n \geq 1

where $f_{n}^{'} (x) = \frac{d f _{n} ( x )}{d x}$

Find the value of

n = 1 \sum 20 f_{n} (x)